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1
In the 8-puzzle game, you are given a grid with numbered square tiles arranged randomly and one tile missing. The goal is to arrange the tiles in descending order by sliding them into the empty space. The tiles can move in 4 directions: left, right, up, and down. Given the initial state of the puzzle below, where the empty spot is represented as “_”, provide the shortest list of tiles that need to be swapped with the empty spot to achieve the goal state. The goal state is when all tiles are in descending order, with the largest number in the top left corner, and the empty spot is in the bottom right corner. The solution should be a list of numbers in Python format, where each number represents the number on the tile that the empty spot is swapped with at each turn. Initial state of the puzzle: [[12, 55, '_'], [35, 97, 43], [17, 25, 9]]
8_puzzle
puzzle
2
[55, 97, 35, 12, 97, 55, 43, 9, 25, 17, 12, 35, 9, 25, 17, 9, 25, 17]
18
0.04202771186828613
18
4
9
[[[12, 55, "_"], [35, 97, 43], [17, 25, 9]]]
[[[12, 55, "_"], [35, 97, 43], [17, 25, 9]]]
["[[12, 55, '_'], [35, 97, 43], [17, 25, 9]]"]
1
In the game 'Sort the Chars', we are given a table of n by m dimensions. This table contains n words, each with m characters, except for the first word which has m - 1 characters. Each character is written on a separate tile. The objective of the game is to rearrange the characters such that row i spells the i-th word in the list, with the blank tile ('_') placed in the top left corner of the board in the end. We can rearrange the tiles by swapping the blank space with any of its 4 diagonal neighboring tiles. Given the list of words and initial state of the board below, where the black space is represented as '_', what is the shortest list of swap actions (reported in python syntax) that can sort the board into the given list of target words? The list must only include the 4 diagonal swap directions: up-right, down-right, up-left, or down-left, representing the direction in ehich the blank space was swpped in. Target words: cam, hill, pray, doer The initial board: [['i', 'c', 'a', 'm'], ['h', 'p', 'l', 'o'], ['_', 'r', 'a', 'y'], ['d', 'l', 'e', 'r']]
8_puzzle_words
puzzle
1
["up-right", "down-right", "down-left", "up-left", "up-right", "down-right", "up-right", "up-left", "down-left", "down-left", "down-right", "up-right", "up-right", "up-left", "down-left", "up-left"]
16
0.21660590171813965
16
4
16
[[["i", "c", "a", "m"], ["h", "p", "l", "o"], ["_", "r", "a", "y"], ["d", "l", "e", "r"]]]
[[["i", "c", "a", "m"], ["h", "p", "l", "o"], ["_", "r", "a", "y"], ["d", "l", "e", "r"]], ["cam", "hill", "pray", "doer"]]
["[['i', 'c', 'a', 'm'], ['h', 'p', 'l', 'o'], ['_', 'r', 'a', 'y'], ['d', 'l', 'e', 'r']]", "['cam', 'hill', 'pray', 'doer']"]
1
We have a map of cities, each represented by a letter, and they are connected by one-way roads. The adjacency matrix below shows the connections between the cities. Each row and column represents a city, and a '1' signifies a direct road from the city of the row to the city of the column. The travel time between any two directly connected cities is the same. Currently, we are located in city 'W'. Our task is to visit city A and city H excatly twice. Determine the quickest route that allows us to visit both these destination cities, ensuring that we stop at the two destinations twice on our path. The sequence in which we visit the destination cities is not important. However, apart from H and A, we can only visit each city once on our path. Provide the solution as a list of the city names on our path, including the start, in Python syntax. G D X N Y W A P V H G 0 0 0 0 0 0 0 0 0 1 D 0 0 0 0 0 0 0 1 0 0 X 1 0 0 0 1 1 0 0 0 0 N 0 0 0 0 0 1 1 0 0 1 Y 0 0 1 1 0 0 1 0 0 0 W 0 0 1 0 0 0 0 0 0 0 A 1 1 1 0 0 0 0 0 1 0 P 1 0 0 0 1 0 0 0 0 1 V 1 0 0 0 0 0 0 1 0 0 H 1 1 0 1 0 0 1 1 0 0
city_directed_graph
pathfinding
10
["W", "X", "G", "H", "A", "D", "P", "H", "A"]
9
0.020942211151123047
9
10
13
[[[0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 1, 0, 0, 1], [0, 0, 1, 1, 0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 0, 0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 1, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 1, 0, 0], [1, 1, 0, 1, 0, 0, 1, 1, 0, 0]], ["G", "D", "X", "N", "Y", "W", "A", "P", "V", "H"], "A", "H"]
[[[0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 1, 0, 0, 1], [0, 0, 1, 1, 0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 0, 0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 1, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 1, 0, 0], [1, 1, 0, 1, 0, 0, 1, 1, 0, 0]], ["G", "D", "X", "N", "Y", "W", "A", "P", "V", "H"], "W", "A", "H"]
["[[0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 1, 0, 0, 1], [0, 0, 1, 1, 0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 0, 0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 1, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 1, 0, 0], [1, 1, 0, 1, 0, 0, 1, 1, 0, 0]]", "['G', 'D', 'X', 'N', 'Y', 'W', 'A', 'P', 'V', 'H']", "['W']", "['A', 'H']"]
1
In the 'taxed coin exchange' problem, you are required to choose a subset of coins from this list [14, 2, 5, 8, 65, 5, 16, 2, 5, 2, 12, 17, 16, 15, 20, 4, 19, 9, 10, 64, 2, 10, 14, 2, 17], such that the sum of the chosen coins adds up to 208. Each coin in the list is unique and can only be used once. Also coins carry a tax value. The tax values for each coin is {20: 13, 4: 2, 17: 16, 19: 8, 64: 4, 5: 3, 14: 10, 10: 9, 65: 11, 16: 2, 12: 7, 8: 5, 2: 2, 15: 14, 9: 8}, where the tax for coins of the same value is the same. Also, if the coin chosen is smaller than the previous one, it must have an even value, otherwise, if the coin is larger than or equal to the previous coin chosen, it must have an odd value. The objective is to determine which subset of coins should be selected to minimize the total tax paid. The solution should be presented as a list of numbers, representing the value of the coins chosen in order, with the first coins chosen being in index 0, formatted in Python syntax.
coin_exchange
subset_sum
15
[64, 16, 4, 5, 5, 19, 65, 16, 12, 2]
44
0.03430962562561035
10
25
25
[[14, 2, 5, 8, 65, 5, 16, 2, 5, 2, 12, 17, 16, 15, 20, 4, 19, 9, 10, 64, 2, 10, 14, 2, 17]]
[[14, 2, 5, 8, 65, 5, 16, 2, 5, 2, 12, 17, 16, 15, 20, 4, 19, 9, 10, 64, 2, 10, 14, 2, 17], {"20": 13, "4": 2, "17": 16, "19": 8, "64": 4, "5": 3, "14": 10, "10": 9, "65": 11, "16": 2, "12": 7, "8": 5, "2": 2, "15": 14, "9": 8}, 208]
["[14, 2, 5, 8, 65, 5, 16, 2, 5, 2, 12, 17, 16, 15, 20, 4, 19, 9, 10, 64, 2, 10, 14, 2, 17]", "{20: 13, 4: 2, 17: 16, 19: 8, 64: 4, 5: 3, 14: 10, 10: 9, 65: 11, 16: 2, 12: 7, 8: 5, 2: 2, 15: 14, 9: 8}", "208"]
1
The game of 'Sort It' begins with 3 tubes, each filled with 4 balls of different colors. The goal is to sort the balls by color, with each tube containing balls of only one color. Only one ball can be moved at a time, taken from the top of one tube and placed on top of another. The capacity of each tube (maximum number of balls we can fit in each tube) is 6 balls. It is not allowed to place a ball in a tube that already has 6 balls. The solution should be a list of tuples, each containing, first, the index of the tube from which a ball is taken and, second, the index of the tube to which it is moved, indexing from 0. Given the initial state of the tubes, represented by the lists below (with the leftmost item being the color of the topmost ball in each tube), what is the shortest list of move tuples that will result in all the balls being correctly sorted? [['Green', 'Red', 'Green', 'Red'], ['Blue', 'Blue', 'Red', 'Green'], ['Red', 'Blue', 'Green', 'Blue']]
color_sorting
sorting
7
[[0, 1], [0, 2], [0, 1], [2, 0], [2, 0], [1, 0], [1, 0], [2, 1], [2, 0], [1, 2], [1, 2], [1, 2], [1, 2], [0, 1], [0, 1], [0, 1], [2, 0]]
17
0.9051551818847656
17
6
12
[[["Green", "Red", "Green", "Red"], ["Blue", "Blue", "Red", "Green"], ["Red", "Blue", "Green", "Blue"]], 6]
[[["Green", "Red", "Green", "Red"], ["Blue", "Blue", "Red", "Green"], ["Red", "Blue", "Green", "Blue"]], 6]
["[['Green', 'Red', 'Green', 'Red'], ['Blue', 'Blue', 'Red', 'Green'], ['Red', 'Blue', 'Green', 'Blue']]", "6"]
1
We have a 3x3 numerical grid, with numbers ranging from 13 to 46 (13 included in the range but 46 is not included). The numbers in each row and column must be strictly increasing or decreasing. This means that either first > second > third or first < second < third in each row and column. If a grid cell is marked with an 'x', the number in that position is hidden. The objective is to replace the 'x's with unique integers from the given range, ensuring that each number only appears once in the grid. The replacements must maintain the consecutive order in each row and column. Additionally, the sum of the numbers in the topmost row plus the numbers in the rightmost column plus the numbers in the diagonal connecting the top-left corner of the grid to its bottom-right corner should be minimized. The solution should be given as a list of tuples in Python syntax. Each tuple should represent the replacement of a number with an 'x' number and contain three elements: the row index of the 'x', the column index of the 'x' (both starting from 0), and the value of the number that replaces the 'x'. The initial state of the grid is as follows: Grid: [['16' 'x' '41'] ['x' '30' 'x'] ['x' '29' '30']]
consecutive_grid
underdetermined_system
7
[[0, 1, 31], [1, 0, 14], [1, 2, 32], [2, 0, 13]]
267
0.23554539680480957
4
33
9
["[['16', '', '41'], ['', '30', ''], ['', '29', '30']]", 13, 46]
["[['16', '', '41'], ['', '30', ''], ['', '29', '30']]", 13, 46]
["[['16', '', '41'], ['', '30', ''], ['', '29', '30']]", "13", "46"]
1
In the magic square problem, a 3x3 grid is filled with unique integers ranging from 29 to 48. Some numbers are already given, while others are unknown and represented as 'x'. Sum of column 1 (counting from 0) must be 107, and sum of row 1 must be 103. Also, the sum of the numbers in the diagonal from the top right to the bottom left corner of the grid should equal 124. The goal is to find unique integers (ie each number can be in the final grid only once) in the given range to replace with ‘x’s in the grid below such that the sum of the specified rows, columns, and diagonal equals the given amounts and the sum of all of the numbers in the grid is as low as possible. The solution should be provided as a list of tuples in Python syntax. Each tuple should contain three numbers for each 'x' position: the row index, the column index (both starting from 0), and the value of the unique integer replaced with 'x' at that position. Grid: [['35' 'x' '46'] ['x' 'x' 'x'] ['x' 'x' 'x']]
magic_square
underdetermined_system
6
[[0, 1, 31], [1, 0, 29], [1, 1, 44], [1, 2, 30], [2, 0, 34], [2, 1, 32], [2, 2, 33]]
314
0.3927474021911621
7
19
9
["[['35', '', '46'], ['', '', ''], ['', '', '']]", 3, 29, 48]
["[['35', '', '46'], ['', '', ''], ['', '', '']]", 29, 48, [1, 2], [1, 2], [107], [103], 124]
["[['35', '', '46'], ['', '', ''], ['', '', '']]", "29", "48", "[None, 107, None]", "[None, 103, None]", "124"]
1
In 'Restricted Sorting', there are 6 stacks each with a capacity of 4 blocks, with 3 stacks filled with blocks of varying shades and the remaining are empty. The challenge is to sort the blocks by shade such that if a stack is not empty, it is stacked with 3 blocks of a single shade. The player can only transfer one block at a time from the top of a stack to an empty stack or to a stack that has only blocks of that shade, without exceeding the stacks’ capacity. Transferring blocks to certain stacks is more expensive than others. The cost of moving one block to the top of each stack is: {0: 5, 1: 5, 2: 3, 3: 1, 4: 5, 5: 1}, where the keys are the index of each stack, indexing from 0. The cost of moving a block is always at least 1. The solution should be a list of tuples, each containing, first, the index of the stack from which a block is picked up from and, second, the index of the stack to which it is transferred, indexing from 0. Given the initial state of the stacks, represented by the lists below (with the leftmost item being the shade of the topmost block in each stack)(and the first stack being the stack at index 0), what is the list of transfer pairs (reported in python syntax) with the least possible cost, that will result in all the blocks being correctly sorted? [[], [], [], ['Blue', 'Yellow', 'Blue', 'Blue'], ['Red', 'Green', 'Yellow', 'Green'], ['Red', 'Red', 'Yellow', 'Green']]
restricted_sorting
sorting
1
[[4, 0], [5, 0], [5, 0], [3, 1], [3, 2], [5, 2], [4, 5], [4, 2], [1, 3], [4, 5]]
32
0.06039690971374512
10
30
12
[[[], [], [], ["Blue", "Yellow", "Blue", "Blue"], ["Red", "Green", "Yellow", "Green"], ["Red", "Red", "Yellow", "Green"]], 4, {"0": 5, "1": 5, "2": 3, "3": 1, "4": 5, "5": 1}]
[[[], [], [], ["Blue", "Yellow", "Blue", "Blue"], ["Red", "Green", "Yellow", "Green"], ["Red", "Red", "Yellow", "Green"]], 4, {"0": 5, "1": 5, "2": 3, "3": 1, "4": 5, "5": 1}, 3]
["[[], [], [], ['Blue', 'Yellow', 'Blue', 'Blue'], ['Red', 'Green', 'Yellow', 'Green'], ['Red', 'Red', 'Yellow', 'Green']]", "{0: 5, 1: 5, 2: 3, 3: 1, 4: 5, 5: 1}", "4", "3"]
1
Using the provided matrix map of a city, where numbers represent travel time in minutes (all numbers are positive integers) and 'x' marks closed workshops, find the quickest route for Ben to travel from his current workshop at index (5, 7) to his destination workshop at index (2, 1), indexing from 0. Ben's car can move north, south, east, or west from a given crossroad, provided there's no x in that direction. Also, there are 3 districts in the city with district 1 covering rows 0 to 2, district 2 covering rows 3 to 4, and district 3 covering rows 5 to 8. Ben has to visit at least 1 workshop in each district on his path to the destination. The roads are bidirectional. The answer should be a list of tuples (in Python syntax) indicating the index of workshops on Ben's path. The start and end workshops must be included in the path. [13 x x 3 10 20 3 x 5] [x x x x 20 16 x x 17] [3 14 18 8 1 20 14 x 7] [13 3 6 10 7 4 6 6 1] [10 12 2 x 11 x 10 8 11] [x x x 11 6 18 13 20 17] [x x 16 x 4 17 7 10 15] [x 7 16 6 19 4 7 x x] [x 11 18 x x x 3 8 x]
traffic
pathfinding
1
[[5, 7], [4, 7], [3, 7], [3, 6], [3, 5], [3, 4], [3, 3], [3, 2], [3, 1], [2, 1]]
64
0.030263185501098633
10
4
4
[[["13", "x", "x", "3", "10", "20", "3", "x", "5"], ["x", "x", "x", "x", "20", "16", "x", "x", "17"], ["3", "14", "18", "8", "1", "20", "14", "x", "7"], ["13", "3", "6", "10", "7", "4", "6", "6", "1"], ["10", "12", "2", "x", "11", "x", "10", "8", "11"], ["x", "x", "x", "11", "6", "18", "13", "20", "17"], ["x", "x", "16", "x", "4", "17", "7", "10", "15"], ["x", "7", "16", "6", "19", "4", "7", "x", "x"], ["x", "11", "18", "x", "x", "x", "3", "8", "x"]]]
[[["13", "x", "x", "3", "10", "20", "3", "x", "5"], ["x", "x", "x", "x", "20", "16", "x", "x", "17"], ["3", "14", "18", "8", "1", "20", "14", "x", "7"], ["13", "3", "6", "10", "7", "4", "6", "6", "1"], ["10", "12", "2", "x", "11", "x", "10", "8", "11"], ["x", "x", "x", "11", "6", "18", "13", "20", "17"], ["x", "x", "16", "x", "4", "17", "7", "10", "15"], ["x", "7", "16", "6", "19", "4", "7", "x", "x"], ["x", "11", "18", "x", "x", "x", "3", "8", "x"]], [5, 7], [2, 1], 2, 4]
["[['13', 'x', 'x', '3', '10', '20', '3', 'x', '5'], ['x', 'x', 'x', 'x', '20', '16', 'x', 'x', '17'], ['3', '14', '18', '8', '1', '20', '14', 'x', '7'], ['13', '3', '6', '10', '7', '4', '6', '6', '1'], ['10', '12', '2', 'x', '11', 'x', '10', '8', '11'], ['x', 'x', 'x', '11', '6', '18', '13', '20', '17'], ['x', 'x', '16', 'x', '4', '17', '7', '10', '15'], ['x', '7', '16', '6', '19', '4', '7', 'x', 'x'], ['x', '11', '18', 'x', 'x', 'x', '3', '8', 'x']]", "(5, 7)", "(2, 1)", "2", "4"]
1
Alex is at a trampoline park with a grid of mini trampolines, arranged in a square of 9x9. Some trampolines are broken and unusable. A map of the park is provided below, with 1 indicating a broken trampoline and 0 indicating a functional one. Alex can jump to any of the eight adjacent trampolines, as long as they are not broken. However, Alex must make excatly 3 diagonal jumps, no more, no less, on his path to his destination. He is currently on the trampoline at position (8, 0) (positions are counted from 0, left to right, top to bottom) and wants to reach the trampoline at position (2, 7). What is the shortest sequence of trampolines he should jump on to reach his destination (including the first and final trampolines)? The answer should be a list of tuples, in Python syntax, indicating the row and column of each trampoline Alex jumps on. 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 0 0 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0
trampoline_matrix
pathfinding
9
[[8, 0], [7, 1], [6, 1], [5, 1], [4, 1], [4, 2], [4, 3], [4, 4], [3, 5], [3, 6], [2, 7]]
11
0.03385114669799805
11
8
2
["[[0, 1, 0, 1, 0, 1, 1, 1, 1], [0, 0, 1, 1, 0, 0, 0, 0, 1], [0, 0, 0, 1, 1, 1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 1, 1, 1, 1, 0, 1, 1], [0, 0, 0, 1, 1, 1, 1, 1, 0], [1, 0, 1, 1, 1, 1, 1, 0, 1], [0, 1, 1, 1, 1, 1, 0, 1, 0]]", 3]
["[[0, 1, 0, 1, 0, 1, 1, 1, 1], [0, 0, 1, 1, 0, 0, 0, 0, 1], [0, 0, 0, 1, 1, 1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 1, 1, 1, 1, 0, 1, 1], [0, 0, 0, 1, 1, 1, 1, 1, 0], [1, 0, 1, 1, 1, 1, 1, 0, 1], [0, 1, 1, 1, 1, 1, 0, 1, 0]]", [8, 0], [2, 7], 3]
["[[0, 1, 0, 1, 0, 1, 1, 1, 1], [0, 0, 1, 1, 0, 0, 0, 0, 1], [0, 0, 0, 1, 1, 1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 1, 1, 1, 1, 0, 1, 1], [0, 0, 0, 1, 1, 1, 1, 1, 0], [1, 0, 1, 1, 1, 1, 1, 0, 1], [0, 1, 1, 1, 1, 1, 0, 1, 0]]", "(8, 0)", "(2, 7)", "3"]
1
Given 5 labeled water jugs with capacities 41, 26, 44, 75, 40, 136 liters, we aim to fill 3 unlabeled buckets, numbered 1 to 3 and arranged in a line in ascending order, with 274, 297, 343 liters of water respectively. The amount of water in each unlabeled bucket can not at any point in time exceed the amount of water in the bucket placed after it. Jugs can only be filled to the top and emptied completely, and the unlabeled buckets cannot be overfilled. An action, represented as a tuple ('+', X, Y) or ('-', X, Y), involves adding to or removing water from the unlabeled bucket numbered Y, using the jug with capacity X. Determine the shortest sequence of actions needed to fill the buckets as specified, and present the solution as a list of action tuples in Python syntax.
water_jug
subset_sum
4
[["+", 40, 3], ["+", 75, 3], ["+", 136, 3], ["-", 44, 3], ["+", 136, 3], ["+", 26, 2], ["+", 40, 2], ["+", 136, 2], ["-", 41, 2], ["+", 136, 2], ["+", 40, 1], ["+", 40, 1], ["+", 44, 1], ["+", 75, 1], ["+", 75, 1]]
15
0.05471658706665039
15
36
3
[[41, 26, 44, 75, 40, 136], [274, 297, 343]]
[[41, 26, 44, 75, 40, 136], [274, 297, 343]]
["[41, 26, 44, 75, 40, 136]", "[274, 297, 343]"]
2
In the 8-puzzle game, you are given a grid with numbered square tiles arranged randomly and one tile missing. The goal is to arrange the tiles in descending order by sliding them into the empty space. The tiles can move in 4 directions: left, right, up, and down. Given the initial state of the puzzle below, where the empty spot is represented as “_”, provide the shortest list of tiles that need to be swapped with the empty spot to achieve the goal state. The goal state is when all tiles are in descending order, with the largest number in the top left corner, and the empty spot is in the bottom right corner. The solution should be a list of numbers in Python format, where each number represents the number on the tile that the empty spot is swapped with at each turn. Initial state of the puzzle: [[99, 69, 19], [53, '_', 30], [100, 63, 40]]
8_puzzle
puzzle
2
[53, 100, 63, 40, 30, 19, 69, 53, 40, 30, 19, 40, 53, 99, 100, 63, 30, 19]
18
0.03798794746398926
18
4
9
[[[99, 69, 19], [53, "_", 30], [100, 63, 40]]]
[[[99, 69, 19], [53, "_", 30], [100, 63, 40]]]
["[[99, 69, 19], [53, '_', 30], [100, 63, 40]]"]
2
In the game 'Sort the Chars', we are given a table of n by m dimensions. This table contains n words, each with m characters, except for the first word which has m - 1 characters. Each character is written on a separate tile. The objective of the game is to rearrange the characters such that row i spells the i-th word in the list, with the blank tile ('_') placed in the top left corner of the board in the end. We can rearrange the tiles by swapping the blank space with any of its 4 diagonal neighboring tiles. Given the list of words and initial state of the board below, where the black space is represented as '_', what is the shortest list of swap actions (reported in python syntax) that can sort the board into the given list of target words? The list must only include the 4 diagonal swap directions: up-right, down-right, up-left, or down-left, representing the direction in ehich the blank space was swpped in. Target words: ako, pole, raff, mudd The initial board: [['o', 'a', 'u', 'o'], ['p', 'r', 'l', 'e'], ['f', 'a', '_', 'f'], ['m', 'k', 'd', 'd']]
8_puzzle_words
puzzle
1
["up-left", "down-left", "down-right", "up-right", "up-left", "up-right", "down-right", "down-left", "up-left", "down-left", "down-right", "up-right", "up-right", "up-left", "down-left", "up-left"]
16
0.23615717887878418
16
4
16
[[["o", "a", "u", "o"], ["p", "r", "l", "e"], ["f", "a", "_", "f"], ["m", "k", "d", "d"]]]
[[["o", "a", "u", "o"], ["p", "r", "l", "e"], ["f", "a", "_", "f"], ["m", "k", "d", "d"]], ["ako", "pole", "raff", "mudd"]]
["[['o', 'a', 'u', 'o'], ['p', 'r', 'l', 'e'], ['f', 'a', '_', 'f'], ['m', 'k', 'd', 'd']]", "['ako', 'pole', 'raff', 'mudd']"]
2
We have a map of cities, each represented by a letter, and they are connected by one-way roads. The adjacency matrix below shows the connections between the cities. Each row and column represents a city, and a '1' signifies a direct road from the city of the row to the city of the column. The travel time between any two directly connected cities is the same. Currently, we are located in city 'D'. Our task is to visit city K and city Q excatly twice. Determine the quickest route that allows us to visit both these destination cities, ensuring that we stop at the two destinations twice on our path. The sequence in which we visit the destination cities is not important. However, apart from Q and K, we can only visit each city once on our path. Provide the solution as a list of the city names on our path, including the start, in Python syntax. S U X D I Z R M K Q S 0 0 0 0 1 0 0 0 0 1 U 0 0 1 0 0 0 0 0 0 0 X 0 0 0 0 0 0 1 0 1 0 D 0 1 0 0 0 1 0 0 0 0 I 0 1 0 0 0 0 0 0 0 1 Z 0 0 0 0 1 0 0 0 1 0 R 1 1 0 0 0 0 0 0 0 0 M 0 0 1 0 1 0 0 0 0 0 K 0 0 0 0 0 0 1 1 0 0 Q 0 1 0 1 1 1 0 0 1 0
city_directed_graph
pathfinding
10
["D", "Z", "I", "Q", "K", "R", "S", "Q", "K"]
9
0.027526378631591797
9
10
13
[[[0, 0, 0, 0, 1, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 1, 0, 0, 0, 1, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 1, 0, 0], [0, 1, 0, 1, 1, 1, 0, 0, 1, 0]], ["S", "U", "X", "D", "I", "Z", "R", "M", "K", "Q"], "K", "Q"]
[[[0, 0, 0, 0, 1, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 1, 0, 0, 0, 1, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 1, 0, 0], [0, 1, 0, 1, 1, 1, 0, 0, 1, 0]], ["S", "U", "X", "D", "I", "Z", "R", "M", "K", "Q"], "D", "K", "Q"]
["[[0, 0, 0, 0, 1, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 1, 0, 0, 0, 1, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 1, 0, 0], [0, 1, 0, 1, 1, 1, 0, 0, 1, 0]]", "['S', 'U', 'X', 'D', 'I', 'Z', 'R', 'M', 'K', 'Q']", "['D']", "['K', 'Q']"]
2
In the 'taxed coin exchange' problem, you are required to choose a subset of coins from this list [13, 10, 12, 10, 16, 7, 9, 14, 13, 18, 36, 11, 7, 14, 7, 8, 19, 7, 39, 13, 3, 18, 20, 4, 3, 18, 15, 2, 19, 16], such that the sum of the chosen coins adds up to 203. Each coin in the list is unique and can only be used once. Also coins carry a tax value. The tax values for each coin is {15: 5, 36: 15, 39: 9, 9: 8, 13: 3, 2: 2, 18: 10, 12: 12, 3: 3, 8: 8, 11: 6, 10: 4, 4: 4, 14: 10, 7: 3, 16: 16, 20: 18, 19: 17}, where the tax for coins of the same value is the same. Also, if the coin chosen is smaller than the previous one, it must have an even value, otherwise, if the coin is larger than or equal to the previous coin chosen, it must have an odd value. The objective is to determine which subset of coins should be selected to minimize the total tax paid. The solution should be presented as a list of numbers, representing the value of the coins chosen in order, with the first coins chosen being in index 0, formatted in Python syntax.
coin_exchange
subset_sum
16
[7, 7, 7, 13, 13, 39, 36, 10, 15, 10, 11, 13, 2, 7, 4, 9]
78
0.03766942024230957
16
30
30
[[13, 10, 12, 10, 16, 7, 9, 14, 13, 18, 36, 11, 7, 14, 7, 8, 19, 7, 39, 13, 3, 18, 20, 4, 3, 18, 15, 2, 19, 16]]
[[13, 10, 12, 10, 16, 7, 9, 14, 13, 18, 36, 11, 7, 14, 7, 8, 19, 7, 39, 13, 3, 18, 20, 4, 3, 18, 15, 2, 19, 16], {"15": 5, "36": 15, "39": 9, "9": 8, "13": 3, "2": 2, "18": 10, "12": 12, "3": 3, "8": 8, "11": 6, "10": 4, "4": 4, "14": 10, "7": 3, "16": 16, "20": 18, "19": 17}, 203]
["[13, 10, 12, 10, 16, 7, 9, 14, 13, 18, 36, 11, 7, 14, 7, 8, 19, 7, 39, 13, 3, 18, 20, 4, 3, 18, 15, 2, 19, 16]", "{15: 5, 36: 15, 39: 9, 9: 8, 13: 3, 2: 2, 18: 10, 12: 12, 3: 3, 8: 8, 11: 6, 10: 4, 4: 4, 14: 10, 7: 3, 16: 16, 20: 18, 19: 17}", "203"]
2
The game of 'Sort It' begins with 3 tubes, each filled with 4 balls of different colors. The goal is to sort the balls by color, with each tube containing balls of only one color. Only one ball can be moved at a time, taken from the top of one tube and placed on top of another. The capacity of each tube (maximum number of balls we can fit in each tube) is 6 balls. It is not allowed to place a ball in a tube that already has 6 balls. The solution should be a list of tuples, each containing, first, the index of the tube from which a ball is taken and, second, the index of the tube to which it is moved, indexing from 0. Given the initial state of the tubes, represented by the lists below (with the leftmost item being the color of the topmost ball in each tube), what is the shortest list of move tuples that will result in all the balls being correctly sorted? [['Red', 'Blue', 'Blue', 'Blue'], ['Green', 'Green', 'Red', 'Green'], ['Green', 'Blue', 'Red', 'Red']]
color_sorting
sorting
7
[[1, 2], [1, 2], [1, 0], [2, 1], [2, 1], [2, 1], [2, 1], [0, 2], [0, 2], [1, 0]]
10
0.05974388122558594
10
6
12
[[["Red", "Blue", "Blue", "Blue"], ["Green", "Green", "Red", "Green"], ["Green", "Blue", "Red", "Red"]], 6]
[[["Red", "Blue", "Blue", "Blue"], ["Green", "Green", "Red", "Green"], ["Green", "Blue", "Red", "Red"]], 6]
["[['Red', 'Blue', 'Blue', 'Blue'], ['Green', 'Green', 'Red', 'Green'], ['Green', 'Blue', 'Red', 'Red']]", "6"]
2
We have a 3x3 numerical grid, with numbers ranging from 8 to 41 (8 included in the range but 41 is not included). The numbers in each row and column must be strictly increasing or decreasing. This means that either first > second > third or first < second < third in each row and column. If a grid cell is marked with an 'x', the number in that position is hidden. The objective is to replace the 'x's with unique integers from the given range, ensuring that each number only appears once in the grid. The replacements must maintain the consecutive order in each row and column. Additionally, the sum of the numbers in the topmost row plus the numbers in the rightmost column plus the numbers in the diagonal connecting the top-left corner of the grid to its bottom-right corner should be minimized. The solution should be given as a list of tuples in Python syntax. Each tuple should represent the replacement of a number with an 'x' number and contain three elements: the row index of the 'x', the column index of the 'x' (both starting from 0), and the value of the number that replaces the 'x'. The initial state of the grid is as follows: Grid: [['10' '15' '39'] ['x' '27' '29'] ['x' 'x' 'x']]
consecutive_grid
underdetermined_system
7
[[1, 0, 11], [2, 0, 30], [2, 1, 28], [2, 2, 8]]
185
0.6896023750305176
4
33
9
["[['10', '15', '39'], ['', '27', '29'], ['', '', '']]", 8, 41]
["[['10', '15', '39'], ['', '27', '29'], ['', '', '']]", 8, 41]
["[['10', '15', '39'], ['', '27', '29'], ['', '', '']]", "8", "41"]
2
In the magic square problem, a 3x3 grid is filled with unique integers ranging from 20 to 39. Some numbers are already given, while others are unknown and represented as 'x'. Sum of column 1 (counting from 0) must be 82, and sum of row 1 must be 87. Also, the sum of the numbers in the diagonal from the top right to the bottom left corner of the grid should equal 94. The goal is to find unique integers (ie each number can be in the final grid only once) in the given range to replace with ‘x’s in the grid below such that the sum of the specified rows, columns, and diagonal equals the given amounts and the sum of all of the numbers in the grid is as low as possible. The solution should be provided as a list of tuples in Python syntax. Each tuple should contain three numbers for each 'x' position: the row index, the column index (both starting from 0), and the value of the unique integer replaced with 'x' at that position. Grid: [['x' 'x' 'x'] ['30' 'x' 'x'] ['x' 'x' 'x']]
magic_square
underdetermined_system
6
[[0, 0, 20], [0, 1, 22], [0, 2, 25], [1, 1, 36], [1, 2, 21], [2, 0, 33], [2, 1, 24], [2, 2, 23]]
234
3.1459109783172607
8
19
9
["[['', '', ''], ['30', '', ''], ['', '', '']]", 3, 20, 39]
["[['', '', ''], ['30', '', ''], ['', '', '']]", 20, 39, [1, 2], [1, 2], [82], [87], 94]
["[['', '', ''], ['30', '', ''], ['', '', '']]", "20", "39", "[None, 82, None]", "[None, 87, None]", "94"]
2
In 'Restricted Sorting', there are 6 stacks each with a capacity of 4 blocks, with 3 stacks filled with blocks of varying shades and the remaining are empty. The challenge is to sort the blocks by shade such that if a stack is not empty, it is stacked with 3 blocks of a single shade. The player can only transfer one block at a time from the top of a stack to an empty stack or to a stack that has only blocks of that shade, without exceeding the stacks’ capacity. Transferring blocks to certain stacks is more expensive than others. The cost of moving one block to the top of each stack is: {0: 5, 1: 3, 2: 3, 3: 1, 4: 3, 5: 2}, where the keys are the index of each stack, indexing from 0. The cost of moving a block is always at least 1. The solution should be a list of tuples, each containing, first, the index of the stack from which a block is picked up from and, second, the index of the stack to which it is transferred, indexing from 0. Given the initial state of the stacks, represented by the lists below (with the leftmost item being the shade of the topmost block in each stack)(and the first stack being the stack at index 0), what is the list of transfer pairs (reported in python syntax) with the least possible cost, that will result in all the blocks being correctly sorted? [['Red', 'Green', 'Red', 'Blue'], ['Red', 'Yellow', 'Yellow', 'Green'], [], ['Blue', 'Yellow', 'Green', 'Blue'], [], []]
restricted_sorting
sorting
1
[[1, 2], [0, 2], [1, 5], [1, 5], [0, 1], [0, 2], [3, 4], [3, 5], [3, 1], [0, 3], [4, 3]]
26
0.10936236381530762
11
30
12
[[["Red", "Green", "Red", "Blue"], ["Red", "Yellow", "Yellow", "Green"], [], ["Blue", "Yellow", "Green", "Blue"], [], []], 4, {"0": 5, "1": 3, "2": 3, "3": 1, "4": 3, "5": 2}]
[[["Red", "Green", "Red", "Blue"], ["Red", "Yellow", "Yellow", "Green"], [], ["Blue", "Yellow", "Green", "Blue"], [], []], 4, {"0": 5, "1": 3, "2": 3, "3": 1, "4": 3, "5": 2}, 3]
["[['Red', 'Green', 'Red', 'Blue'], ['Red', 'Yellow', 'Yellow', 'Green'], [], ['Blue', 'Yellow', 'Green', 'Blue'], [], []]", "{0: 5, 1: 3, 2: 3, 3: 1, 4: 3, 5: 2}", "4", "3"]
2
Using the provided matrix map of a city, where numbers represent travel time in minutes (all numbers are positive integers) and 'x' marks closed workshops, find the quickest route for Ben to travel from his current workshop at index (2, 2) to his destination workshop at index (5, 8), indexing from 0. Ben's car can move north, south, east, or west from a given crossroad, provided there's no x in that direction. Also, there are 3 districts in the city with district 1 covering rows 0 to 1, district 2 covering rows 2 to 4, and district 3 covering rows 5 to 8. Ben has to visit at least 1 workshop in each district on his path to the destination. The roads are bidirectional. The answer should be a list of tuples (in Python syntax) indicating the index of workshops on Ben's path. The start and end workshops must be included in the path. [x 13 9 x x 16 6 18 7] [x 3 11 x x 5 7 x 19] [8 x 2 5 6 12 9 10 7] [12 1 6 20 19 18 12 x 14] [16 5 10 3 x x x 1 9] [x 3 x x x x x x 4] [7 10 x 1 x x 17 x 8] [x 5 x x x 17 x 9 x] [13 16 8 15 x 2 x 1 2]
traffic
pathfinding
1
[[2, 2], [2, 3], [2, 4], [2, 5], [1, 5], [1, 6], [2, 6], [2, 7], [2, 8], [3, 8], [4, 8], [5, 8]]
88
0.027752399444580078
12
4
4
[[["x", "13", "9", "x", "x", "16", "6", "18", "7"], ["x", "3", "11", "x", "x", "5", "7", "x", "19"], ["8", "x", "2", "5", "6", "12", "9", "10", "7"], ["12", "1", "6", "20", "19", "18", "12", "x", "14"], ["16", "5", "10", "3", "x", "x", "x", "1", "9"], ["x", "3", "x", "x", "x", "x", "x", "x", "4"], ["7", "10", "x", "1", "x", "x", "17", "x", "8"], ["x", "5", "x", "x", "x", "17", "x", "9", "x"], ["13", "16", "8", "15", "x", "2", "x", "1", "2"]]]
[[["x", "13", "9", "x", "x", "16", "6", "18", "7"], ["x", "3", "11", "x", "x", "5", "7", "x", "19"], ["8", "x", "2", "5", "6", "12", "9", "10", "7"], ["12", "1", "6", "20", "19", "18", "12", "x", "14"], ["16", "5", "10", "3", "x", "x", "x", "1", "9"], ["x", "3", "x", "x", "x", "x", "x", "x", "4"], ["7", "10", "x", "1", "x", "x", "17", "x", "8"], ["x", "5", "x", "x", "x", "17", "x", "9", "x"], ["13", "16", "8", "15", "x", "2", "x", "1", "2"]], [2, 2], [5, 8], 1, 4]
["[['x', '13', '9', 'x', 'x', '16', '6', '18', '7'], ['x', '3', '11', 'x', 'x', '5', '7', 'x', '19'], ['8', 'x', '2', '5', '6', '12', '9', '10', '7'], ['12', '1', '6', '20', '19', '18', '12', 'x', '14'], ['16', '5', '10', '3', 'x', 'x', 'x', '1', '9'], ['x', '3', 'x', 'x', 'x', 'x', 'x', 'x', '4'], ['7', '10', 'x', '1', 'x', 'x', '17', 'x', '8'], ['x', '5', 'x', 'x', 'x', '17', 'x', '9', 'x'], ['13', '16', '8', '15', 'x', '2', 'x', '1', '2']]", "(2, 2)", "(5, 8)", "1", "4"]
2
Alex is at a trampoline park with a grid of mini trampolines, arranged in a square of 9x9. Some trampolines are broken and unusable. A map of the park is provided below, with 1 indicating a broken trampoline and 0 indicating a functional one. Alex can jump to any of the eight adjacent trampolines, as long as they are not broken. However, Alex must make excatly 3 diagonal jumps, no more, no less, on his path to his destination. He is currently on the trampoline at position (0, 4) (positions are counted from 0, left to right, top to bottom) and wants to reach the trampoline at position (8, 0). What is the shortest sequence of trampolines he should jump on to reach his destination (including the first and final trampolines)? The answer should be a list of tuples, in Python syntax, indicating the row and column of each trampoline Alex jumps on. 1 1 0 1 0 0 1 1 1 1 1 1 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0
trampoline_matrix
pathfinding
9
[[0, 4], [0, 5], [1, 5], [2, 5], [2, 6], [3, 6], [4, 5], [5, 4], [5, 3], [5, 2], [6, 2], [7, 1], [8, 1], [8, 0]]
14
0.03374505043029785
14
8
2
["[[1, 1, 0, 1, 0, 0, 1, 1, 1], [1, 1, 1, 0, 1, 0, 1, 1, 1], [0, 1, 0, 1, 1, 0, 0, 0, 0], [1, 1, 1, 0, 1, 1, 0, 1, 0], [0, 1, 0, 1, 1, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 1, 0, 1], [0, 1, 0, 1, 1, 0, 1, 0, 0], [1, 0, 0, 1, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0]]", 3]
["[[1, 1, 0, 1, 0, 0, 1, 1, 1], [1, 1, 1, 0, 1, 0, 1, 1, 1], [0, 1, 0, 1, 1, 0, 0, 0, 0], [1, 1, 1, 0, 1, 1, 0, 1, 0], [0, 1, 0, 1, 1, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 1, 0, 1], [0, 1, 0, 1, 1, 0, 1, 0, 0], [1, 0, 0, 1, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0]]", [0, 4], [8, 0], 3]
["[[1, 1, 0, 1, 0, 0, 1, 1, 1], [1, 1, 1, 0, 1, 0, 1, 1, 1], [0, 1, 0, 1, 1, 0, 0, 0, 0], [1, 1, 1, 0, 1, 1, 0, 1, 0], [0, 1, 0, 1, 1, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 1, 0, 1], [0, 1, 0, 1, 1, 0, 1, 0, 0], [1, 0, 0, 1, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0]]", "(0, 4)", "(8, 0)", "3"]
2
Given 5 labeled water jugs with capacities 31, 102, 53, 33, 32, 24 liters, we aim to fill 3 unlabeled buckets, numbered 1 to 3 and arranged in a line in ascending order, with 188, 198, 289 liters of water respectively. The amount of water in each unlabeled bucket can not at any point in time exceed the amount of water in the bucket placed after it. Jugs can only be filled to the top and emptied completely, and the unlabeled buckets cannot be overfilled. An action, represented as a tuple ('+', X, Y) or ('-', X, Y), involves adding to or removing water from the unlabeled bucket numbered Y, using the jug with capacity X. Determine the shortest sequence of actions needed to fill the buckets as specified, and present the solution as a list of action tuples in Python syntax.
water_jug
subset_sum
4
[["+", 32, 3], ["+", 102, 3], ["+", 102, 3], ["+", 53, 3], ["+", 31, 2], ["+", 32, 2], ["+", 33, 2], ["+", 102, 2], ["+", 33, 1], ["+", 53, 1], ["+", 102, 1]]
11
0.04619026184082031
11
36
3
[[31, 102, 53, 33, 32, 24], [188, 198, 289]]
[[31, 102, 53, 33, 32, 24], [188, 198, 289]]
["[31, 102, 53, 33, 32, 24]", "[188, 198, 289]"]
3
In the 8-puzzle game, you are given a grid with numbered square tiles arranged randomly and one tile missing. The goal is to arrange the tiles in descending order by sliding them into the empty space. The tiles can move in 4 directions: left, right, up, and down. Given the initial state of the puzzle below, where the empty spot is represented as “_”, provide the shortest list of tiles that need to be swapped with the empty spot to achieve the goal state. The goal state is when all tiles are in descending order, with the largest number in the top left corner, and the empty spot is in the bottom right corner. The solution should be a list of numbers in Python format, where each number represents the number on the tile that the empty spot is swapped with at each turn. Initial state of the puzzle: [[87, 50, 82], [68, '_', 74], [23, 61, 11]]
8_puzzle
puzzle
2
[50, 82, 74, 50, 61, 11]
6
0.028201580047607422
6
4
9
[[[87, 50, 82], [68, "_", 74], [23, 61, 11]]]
[[[87, 50, 82], [68, "_", 74], [23, 61, 11]]]
["[[87, 50, 82], [68, '_', 74], [23, 61, 11]]"]
3
In the game 'Sort the Chars', we are given a table of n by m dimensions. This table contains n words, each with m characters, except for the first word which has m - 1 characters. Each character is written on a separate tile. The objective of the game is to rearrange the characters such that row i spells the i-th word in the list, with the blank tile ('_') placed in the top left corner of the board in the end. We can rearrange the tiles by swapping the blank space with any of its 4 diagonal neighboring tiles. Given the list of words and initial state of the board below, where the black space is represented as '_', what is the shortest list of swap actions (reported in python syntax) that can sort the board into the given list of target words? The list must only include the 4 diagonal swap directions: up-right, down-right, up-left, or down-left, representing the direction in ehich the blank space was swpped in. Target words: twi, ovey, udic, diem The initial board: [['v', 't', 'u', 'i'], ['o', 'w', 'e', 'i'], ['y', 'd', '_', 'c'], ['d', 'i', 'e', 'm']]
8_puzzle_words
puzzle
1
["up-left", "up-right", "down-right", "down-left", "up-left", "down-left", "down-right", "up-right", "up-left", "up-right", "down-right", "down-left", "up-left", "down-left", "down-right", "up-right", "up-left", "up-left"]
18
0.22588300704956055
18
4
16
[[["v", "t", "u", "i"], ["o", "w", "e", "i"], ["y", "d", "_", "c"], ["d", "i", "e", "m"]]]
[[["v", "t", "u", "i"], ["o", "w", "e", "i"], ["y", "d", "_", "c"], ["d", "i", "e", "m"]], ["twi", "ovey", "udic", "diem"]]
["[['v', 't', 'u', 'i'], ['o', 'w', 'e', 'i'], ['y', 'd', '_', 'c'], ['d', 'i', 'e', 'm']]", "['twi', 'ovey', 'udic', 'diem']"]
3
We have a map of cities, each represented by a letter, and they are connected by one-way roads. The adjacency matrix below shows the connections between the cities. Each row and column represents a city, and a '1' signifies a direct road from the city of the row to the city of the column. The travel time between any two directly connected cities is the same. Currently, we are located in city 'M'. Our task is to visit city A and city R excatly twice. Determine the quickest route that allows us to visit both these destination cities, ensuring that we stop at the two destinations twice on our path. The sequence in which we visit the destination cities is not important. However, apart from R and A, we can only visit each city once on our path. Provide the solution as a list of the city names on our path, including the start, in Python syntax. O T I F A M X Y R G O 0 0 0 0 1 0 0 0 0 0 T 1 0 1 0 0 0 0 0 0 0 I 0 0 0 0 1 1 0 0 1 1 F 0 0 1 0 0 0 0 0 0 1 A 1 1 0 0 0 0 1 0 0 0 M 0 0 0 1 0 0 0 0 0 1 X 0 0 1 1 1 0 0 1 0 0 Y 1 1 0 0 0 0 1 0 1 1 R 1 1 0 1 0 0 0 1 0 0 G 0 0 0 1 1 0 0 0 1 0
city_directed_graph
pathfinding
10
["M", "G", "R", "Y", "R", "O", "A", "X", "A"]
9
0.0287935733795166
9
10
13
[[[0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 1, 0, 0, 1, 1], [0, 0, 1, 0, 0, 0, 0, 0, 0, 1], [1, 1, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0, 0, 1, 0, 0], [1, 1, 0, 0, 0, 0, 1, 0, 1, 1], [1, 1, 0, 1, 0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 1, 0, 0, 0, 1, 0]], ["O", "T", "I", "F", "A", "M", "X", "Y", "R", "G"], "A", "R"]
[[[0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 1, 0, 0, 1, 1], [0, 0, 1, 0, 0, 0, 0, 0, 0, 1], [1, 1, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0, 0, 1, 0, 0], [1, 1, 0, 0, 0, 0, 1, 0, 1, 1], [1, 1, 0, 1, 0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 1, 0, 0, 0, 1, 0]], ["O", "T", "I", "F", "A", "M", "X", "Y", "R", "G"], "M", "A", "R"]
["[[0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 1, 0, 0, 1, 1], [0, 0, 1, 0, 0, 0, 0, 0, 0, 1], [1, 1, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0, 0, 1, 0, 0], [1, 1, 0, 0, 0, 0, 1, 0, 1, 1], [1, 1, 0, 1, 0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 1, 0, 0, 0, 1, 0]]", "['O', 'T', 'I', 'F', 'A', 'M', 'X', 'Y', 'R', 'G']", "['M']", "['A', 'R']"]
3
In the 'taxed coin exchange' problem, you are required to choose a subset of coins from this list [13, 19, 9, 4, 18, 17, 2, 14, 17, 17, 8, 6, 4, 10, 19, 37, 20, 19, 14, 7, 6, 11, 21, 9, 3, 10, 21, 20, 2, 17, 2], such that the sum of the chosen coins adds up to 215. Each coin in the list is unique and can only be used once. Also coins carry a tax value. The tax values for each coin is {14: 2, 18: 9, 11: 10, 9: 6, 10: 9, 13: 10, 37: 9, 6: 5, 2: 2, 4: 1, 8: 3, 19: 10, 17: 7, 7: 6, 3: 3, 21: 3, 20: 4}, where the tax for coins of the same value is the same. Also, if the coin chosen is smaller than the previous one, it must have an even value, otherwise, if the coin is larger than or equal to the previous coin chosen, it must have an odd value. The objective is to determine which subset of coins should be selected to minimize the total tax paid. The solution should be presented as a list of numbers, representing the value of the coins chosen in order, with the first coins chosen being in index 0, formatted in Python syntax.
coin_exchange
subset_sum
17
[14, 21, 20, 4, 37, 4, 21, 20, 18, 8, 17, 14, 17]
55
0.0338587760925293
13
31
31
[[13, 19, 9, 4, 18, 17, 2, 14, 17, 17, 8, 6, 4, 10, 19, 37, 20, 19, 14, 7, 6, 11, 21, 9, 3, 10, 21, 20, 2, 17, 2]]
[[13, 19, 9, 4, 18, 17, 2, 14, 17, 17, 8, 6, 4, 10, 19, 37, 20, 19, 14, 7, 6, 11, 21, 9, 3, 10, 21, 20, 2, 17, 2], {"14": 2, "18": 9, "11": 10, "9": 6, "10": 9, "13": 10, "37": 9, "6": 5, "2": 2, "4": 1, "8": 3, "19": 10, "17": 7, "7": 6, "3": 3, "21": 3, "20": 4}, 215]
["[13, 19, 9, 4, 18, 17, 2, 14, 17, 17, 8, 6, 4, 10, 19, 37, 20, 19, 14, 7, 6, 11, 21, 9, 3, 10, 21, 20, 2, 17, 2]", "{14: 2, 18: 9, 11: 10, 9: 6, 10: 9, 13: 10, 37: 9, 6: 5, 2: 2, 4: 1, 8: 3, 19: 10, 17: 7, 7: 6, 3: 3, 21: 3, 20: 4}", "215"]
3
The game of 'Sort It' begins with 3 tubes, each filled with 4 balls of different colors. The goal is to sort the balls by color, with each tube containing balls of only one color. Only one ball can be moved at a time, taken from the top of one tube and placed on top of another. The capacity of each tube (maximum number of balls we can fit in each tube) is 6 balls. It is not allowed to place a ball in a tube that already has 6 balls. The solution should be a list of tuples, each containing, first, the index of the tube from which a ball is taken and, second, the index of the tube to which it is moved, indexing from 0. Given the initial state of the tubes, represented by the lists below (with the leftmost item being the color of the topmost ball in each tube), what is the shortest list of move tuples that will result in all the balls being correctly sorted? [['Red', 'Blue', 'Blue', 'Red'], ['Red', 'Blue', 'Blue', 'Green'], ['Green', 'Green', 'Red', 'Green']]
color_sorting
sorting
7
[[0, 1], [2, 0], [2, 0], [2, 1], [0, 2], [0, 2], [0, 2], [0, 2], [1, 0], [1, 0], [1, 0], [1, 0], [1, 2], [1, 0], [2, 1], [2, 1], [2, 1], [0, 2], [0, 1]]
19
2.109553337097168
19
6
12
[[["Red", "Blue", "Blue", "Red"], ["Red", "Blue", "Blue", "Green"], ["Green", "Green", "Red", "Green"]], 6]
[[["Red", "Blue", "Blue", "Red"], ["Red", "Blue", "Blue", "Green"], ["Green", "Green", "Red", "Green"]], 6]
["[['Red', 'Blue', 'Blue', 'Red'], ['Red', 'Blue', 'Blue', 'Green'], ['Green', 'Green', 'Red', 'Green']]", "6"]
3
We have a 3x3 numerical grid, with numbers ranging from 20 to 53 (20 included in the range but 53 is not included). The numbers in each row and column must be strictly increasing or decreasing. This means that either first > second > third or first < second < third in each row and column. If a grid cell is marked with an 'x', the number in that position is hidden. The objective is to replace the 'x's with unique integers from the given range, ensuring that each number only appears once in the grid. The replacements must maintain the consecutive order in each row and column. Additionally, the sum of the numbers in the topmost row plus the numbers in the rightmost column plus the numbers in the diagonal connecting the top-left corner of the grid to its bottom-right corner should be minimized. The solution should be given as a list of tuples in Python syntax. Each tuple should represent the replacement of a number with an 'x' number and contain three elements: the row index of the 'x', the column index of the 'x' (both starting from 0), and the value of the number that replaces the 'x'. The initial state of the grid is as follows: Grid: [['22' 'x' '48'] ['27' 'x' 'x'] ['x' '42' '52']]
consecutive_grid
underdetermined_system
7
[[0, 1, 23], [1, 1, 28], [1, 2, 49], [2, 0, 29]]
344
0.18951058387756348
4
33
9
["[['22', '', '48'], ['27', '', ''], ['', '42', '52']]", 20, 53]
["[['22', '', '48'], ['27', '', ''], ['', '42', '52']]", 20, 53]
["[['22', '', '48'], ['27', '', ''], ['', '42', '52']]", "20", "53"]
3
In the magic square problem, a 3x3 grid is filled with unique integers ranging from 20 to 39. Some numbers are already given, while others are unknown and represented as 'x'. Sum of column 1 (counting from 0) must be 99, and sum of row 1 must be 86. Also, the sum of the numbers in the diagonal from the top right to the bottom left corner of the grid should equal 76. The goal is to find unique integers (ie each number can be in the final grid only once) in the given range to replace with ‘x’s in the grid below such that the sum of the specified rows, columns, and diagonal equals the given amounts and the sum of all of the numbers in the grid is as low as possible. The solution should be provided as a list of tuples in Python syntax. Each tuple should contain three numbers for each 'x' position: the row index, the column index (both starting from 0), and the value of the unique integer replaced with 'x' at that position. Grid: [['25' 'x' 'x'] ['x' 'x' 'x'] ['x' 'x' 'x']]
magic_square
underdetermined_system
6
[[0, 1, 26], [0, 2, 20], [1, 0, 23], [1, 1, 35], [1, 2, 28], [2, 0, 21], [2, 1, 38], [2, 2, 22]]
238
6.525880575180054
8
19
9
["[['25', '', ''], ['', '', ''], ['', '', '']]", 3, 20, 39]
["[['25', '', ''], ['', '', ''], ['', '', '']]", 20, 39, [1, 2], [1, 2], [99], [86], 76]
["[['25', '', ''], ['', '', ''], ['', '', '']]", "20", "39", "[None, 99, None]", "[None, 86, None]", "76"]
3
In 'Restricted Sorting', there are 6 stacks each with a capacity of 4 blocks, with 3 stacks filled with blocks of varying shades and the remaining are empty. The challenge is to sort the blocks by shade such that if a stack is not empty, it is stacked with 3 blocks of a single shade. The player can only transfer one block at a time from the top of a stack to an empty stack or to a stack that has only blocks of that shade, without exceeding the stacks’ capacity. Transferring blocks to certain stacks is more expensive than others. The cost of moving one block to the top of each stack is: {0: 3, 1: 6, 2: 5, 3: 6, 4: 1, 5: 1}, where the keys are the index of each stack, indexing from 0. The cost of moving a block is always at least 1. The solution should be a list of tuples, each containing, first, the index of the stack from which a block is picked up from and, second, the index of the stack to which it is transferred, indexing from 0. Given the initial state of the stacks, represented by the lists below (with the leftmost item being the shade of the topmost block in each stack)(and the first stack being the stack at index 0), what is the list of transfer pairs (reported in python syntax) with the least possible cost, that will result in all the blocks being correctly sorted? [[], ['Yellow', 'Green', 'Red', 'Red'], ['Blue', 'Green', 'Red', 'Green'], ['Yellow', 'Blue', 'Yellow', 'Blue'], [], []]
restricted_sorting
sorting
1
[[1, 0], [1, 4], [2, 5], [2, 4], [2, 1], [3, 0], [3, 5], [3, 0], [2, 4], [3, 5]]
21
0.11831974983215332
10
30
12
[[[], ["Yellow", "Green", "Red", "Red"], ["Blue", "Green", "Red", "Green"], ["Yellow", "Blue", "Yellow", "Blue"], [], []], 4, {"0": 3, "1": 6, "2": 5, "3": 6, "4": 1, "5": 1}]
[[[], ["Yellow", "Green", "Red", "Red"], ["Blue", "Green", "Red", "Green"], ["Yellow", "Blue", "Yellow", "Blue"], [], []], 4, {"0": 3, "1": 6, "2": 5, "3": 6, "4": 1, "5": 1}, 3]
["[[], ['Yellow', 'Green', 'Red', 'Red'], ['Blue', 'Green', 'Red', 'Green'], ['Yellow', 'Blue', 'Yellow', 'Blue'], [], []]", "{0: 3, 1: 6, 2: 5, 3: 6, 4: 1, 5: 1}", "4", "3"]
3
Using the provided matrix map of a city, where numbers represent travel time in minutes (all numbers are positive integers) and 'x' marks closed workshops, find the quickest route for Ben to travel from his current workshop at index (7, 7) to his destination workshop at index (1, 2), indexing from 0. Ben's car can move north, south, east, or west from a given crossroad, provided there's no x in that direction. Also, there are 3 districts in the city with district 1 covering rows 0 to 1, district 2 covering rows 2 to 6, and district 3 covering rows 7 to 8. Ben has to visit at least 1 workshop in each district on his path to the destination. The roads are bidirectional. The answer should be a list of tuples (in Python syntax) indicating the index of workshops on Ben's path. The start and end workshops must be included in the path. [10 14 4 16 x 17 5 x x] [5 20 16 3 1 8 x 16 19] [1 x 5 13 3 15 19 15 x] [x 16 13 20 x 8 x x x] [x 16 x 17 11 1 x 15 x] [10 x 14 11 5 7 12 x 5] [7 x 15 x 15 8 3 6 7] [x 12 14 x x x 16 3 13] [19 5 10 x x x 9 7 14]
traffic
pathfinding
1
[[7, 7], [6, 7], [6, 6], [6, 5], [5, 5], [4, 5], [3, 5], [2, 5], [2, 4], [1, 4], [1, 3], [1, 2]]
71
0.02632451057434082
12
4
4
[[["10", "14", "4", "16", "x", "17", "5", "x", "x"], ["5", "20", "16", "3", "1", "8", "x", "16", "19"], ["1", "x", "5", "13", "3", "15", "19", "15", "x"], ["x", "16", "13", "20", "x", "8", "x", "x", "x"], ["x", "16", "x", "17", "11", "1", "x", "15", "x"], ["10", "x", "14", "11", "5", "7", "12", "x", "5"], ["7", "x", "15", "x", "15", "8", "3", "6", "7"], ["x", "12", "14", "x", "x", "x", "16", "3", "13"], ["19", "5", "10", "x", "x", "x", "9", "7", "14"]]]
[[["10", "14", "4", "16", "x", "17", "5", "x", "x"], ["5", "20", "16", "3", "1", "8", "x", "16", "19"], ["1", "x", "5", "13", "3", "15", "19", "15", "x"], ["x", "16", "13", "20", "x", "8", "x", "x", "x"], ["x", "16", "x", "17", "11", "1", "x", "15", "x"], ["10", "x", "14", "11", "5", "7", "12", "x", "5"], ["7", "x", "15", "x", "15", "8", "3", "6", "7"], ["x", "12", "14", "x", "x", "x", "16", "3", "13"], ["19", "5", "10", "x", "x", "x", "9", "7", "14"]], [7, 7], [1, 2], 1, 6]
["[['10', '14', '4', '16', 'x', '17', '5', 'x', 'x'], ['5', '20', '16', '3', '1', '8', 'x', '16', '19'], ['1', 'x', '5', '13', '3', '15', '19', '15', 'x'], ['x', '16', '13', '20', 'x', '8', 'x', 'x', 'x'], ['x', '16', 'x', '17', '11', '1', 'x', '15', 'x'], ['10', 'x', '14', '11', '5', '7', '12', 'x', '5'], ['7', 'x', '15', 'x', '15', '8', '3', '6', '7'], ['x', '12', '14', 'x', 'x', 'x', '16', '3', '13'], ['19', '5', '10', 'x', 'x', 'x', '9', '7', '14']]", "(7, 7)", "(1, 2)", "1", "6"]
3
Alex is at a trampoline park with a grid of mini trampolines, arranged in a square of 9x9. Some trampolines are broken and unusable. A map of the park is provided below, with 1 indicating a broken trampoline and 0 indicating a functional one. Alex can jump to any of the eight adjacent trampolines, as long as they are not broken. However, Alex must make excatly 3 diagonal jumps, no more, no less, on his path to his destination. He is currently on the trampoline at position (8, 6) (positions are counted from 0, left to right, top to bottom) and wants to reach the trampoline at position (0, 0). What is the shortest sequence of trampolines he should jump on to reach his destination (including the first and final trampolines)? The answer should be a list of tuples, in Python syntax, indicating the row and column of each trampoline Alex jumps on. 0 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 1 1 1 0 0 0 0
trampoline_matrix
pathfinding
9
[[8, 6], [7, 5], [6, 5], [5, 5], [4, 6], [3, 5], [3, 4], [2, 4], [2, 3], [2, 2], [1, 2], [1, 1], [1, 0], [0, 0]]
14
0.03221940994262695
14
8
2
["[[0, 1, 1, 0, 1, 1, 1, 1, 0], [0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0, 0, 0, 0], [1, 0, 1, 1, 1, 0, 0, 0, 0], [0, 1, 1, 0, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 0, 0, 0, 0]]", 3]
["[[0, 1, 1, 0, 1, 1, 1, 1, 0], [0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0, 0, 0, 0], [1, 0, 1, 1, 1, 0, 0, 0, 0], [0, 1, 1, 0, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 0, 0, 0, 0]]", [8, 6], [0, 0], 3]
["[[0, 1, 1, 0, 1, 1, 1, 1, 0], [0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 0, 1, 1], [0, 1, 1, 1, 1, 1, 0, 0, 1], [1, 0, 0, 1, 1, 0, 0, 0, 0], [1, 0, 1, 1, 1, 0, 0, 0, 0], [0, 1, 1, 0, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 0, 0, 0, 0]]", "(8, 6)", "(0, 0)", "3"]
3
Given 5 labeled water jugs with capacities 43, 149, 131, 132, 70, 104 liters, we aim to fill 3 unlabeled buckets, numbered 1 to 3 and arranged in a line in ascending order, with 284, 297, 436 liters of water respectively. The amount of water in each unlabeled bucket can not at any point in time exceed the amount of water in the bucket placed after it. Jugs can only be filled to the top and emptied completely, and the unlabeled buckets cannot be overfilled. An action, represented as a tuple ('+', X, Y) or ('-', X, Y), involves adding to or removing water from the unlabeled bucket numbered Y, using the jug with capacity X. Determine the shortest sequence of actions needed to fill the buckets as specified, and present the solution as a list of action tuples in Python syntax.
water_jug
subset_sum
4
[["+", 131, 3], ["+", 131, 3], ["+", 70, 3], ["+", 104, 3], ["+", 131, 2], ["+", 132, 2], ["-", 70, 2], ["+", 104, 2], ["+", 104, 1], ["+", 104, 1], ["-", 132, 1], ["+", 104, 1], ["+", 104, 1]]
13
0.038121938705444336
13
36
3
[[43, 149, 131, 132, 70, 104], [284, 297, 436]]
[[43, 149, 131, 132, 70, 104], [284, 297, 436]]
["[43, 149, 131, 132, 70, 104]", "[284, 297, 436]"]
4
In the 8-puzzle game, you are given a grid with numbered square tiles arranged randomly and one tile missing. The goal is to arrange the tiles in descending order by sliding them into the empty space. The tiles can move in 4 directions: left, right, up, and down. Given the initial state of the puzzle below, where the empty spot is represented as “_”, provide the shortest list of tiles that need to be swapped with the empty spot to achieve the goal state. The goal state is when all tiles are in descending order, with the largest number in the top left corner, and the empty spot is in the bottom right corner. The solution should be a list of numbers in Python format, where each number represents the number on the tile that the empty spot is swapped with at each turn. Initial state of the puzzle: [[66, 89, 70], [25, 42, 83], ['_', 24, 71]]
8_puzzle
puzzle
2
[24, 42, 89, 70, 83, 71, 42, 24, 25, 66, 70, 89, 66, 70, 89, 83, 71, 42]
18
0.03897857666015625
18
4
9
[[[66, 89, 70], [25, 42, 83], ["_", 24, 71]]]
[[[66, 89, 70], [25, 42, 83], ["_", 24, 71]]]
["[[66, 89, 70], [25, 42, 83], ['_', 24, 71]]"]
4
In the game 'Sort the Chars', we are given a table of n by m dimensions. This table contains n words, each with m characters, except for the first word which has m - 1 characters. Each character is written on a separate tile. The objective of the game is to rearrange the characters such that row i spells the i-th word in the list, with the blank tile ('_') placed in the top left corner of the board in the end. We can rearrange the tiles by swapping the blank space with any of its 4 diagonal neighboring tiles. Given the list of words and initial state of the board below, where the black space is represented as '_', what is the shortest list of swap actions (reported in python syntax) that can sort the board into the given list of target words? The list must only include the 4 diagonal swap directions: up-right, down-right, up-left, or down-left, representing the direction in ehich the blank space was swpped in. Target words: map, cere, drub, velo The initial board: [['_', 'm', 'u', 'p'], ['c', 'e', 'r', 'a'], ['d', 'r', 'e', 'b'], ['v', 'e', 'l', 'o']]
8_puzzle_words
puzzle
1
["down-right", "up-right", "down-right", "down-left", "up-left", "up-left"]
6
0.2581043243408203
6
4
16
[[["_", "m", "u", "p"], ["c", "e", "r", "a"], ["d", "r", "e", "b"], ["v", "e", "l", "o"]]]
[[["_", "m", "u", "p"], ["c", "e", "r", "a"], ["d", "r", "e", "b"], ["v", "e", "l", "o"]], ["map", "cere", "drub", "velo"]]
["[['_', 'm', 'u', 'p'], ['c', 'e', 'r', 'a'], ['d', 'r', 'e', 'b'], ['v', 'e', 'l', 'o']]", "['map', 'cere', 'drub', 'velo']"]
4
We have a map of cities, each represented by a letter, and they are connected by one-way roads. The adjacency matrix below shows the connections between the cities. Each row and column represents a city, and a '1' signifies a direct road from the city of the row to the city of the column. The travel time between any two directly connected cities is the same. Currently, we are located in city 'L'. Our task is to visit city D and city N excatly twice. Determine the quickest route that allows us to visit both these destination cities, ensuring that we stop at the two destinations twice on our path. The sequence in which we visit the destination cities is not important. However, apart from N and D, we can only visit each city once on our path. Provide the solution as a list of the city names on our path, including the start, in Python syntax. D E Z R N I K G L B D 0 1 1 1 0 0 0 1 1 1 E 0 0 1 0 0 1 0 0 0 0 Z 0 0 0 0 1 0 0 0 0 0 R 1 0 1 0 0 0 0 0 0 1 N 0 1 0 0 0 1 1 1 0 0 I 1 0 0 0 1 0 1 0 1 0 K 0 0 1 0 1 0 0 0 0 0 G 0 0 1 1 0 1 0 0 0 0 L 0 0 1 0 0 0 0 1 0 0 B 0 1 0 0 0 1 1 0 0 0
city_directed_graph
pathfinding
10
["L", "Z", "N", "K", "N", "I", "D", "R", "D"]
9
0.026905059814453125
9
10
13
[[[0, 1, 1, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 0, 1, 0, 1, 0], [0, 0, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 1, 0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 1, 1, 0, 0, 0]], ["D", "E", "Z", "R", "N", "I", "K", "G", "L", "B"], "D", "N"]
[[[0, 1, 1, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 0, 1, 0, 1, 0], [0, 0, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 1, 0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 1, 1, 0, 0, 0]], ["D", "E", "Z", "R", "N", "I", "K", "G", "L", "B"], "L", "D", "N"]
["[[0, 1, 1, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 1, 1, 1, 0, 0], [1, 0, 0, 0, 1, 0, 1, 0, 1, 0], [0, 0, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 1, 0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 1, 1, 0, 0, 0]]", "['D', 'E', 'Z', 'R', 'N', 'I', 'K', 'G', 'L', 'B']", "['L']", "['D', 'N']"]
4
In the 'taxed coin exchange' problem, you are required to choose a subset of coins from this list [14, 19, 3, 8, 5, 107, 4, 10, 9, 2, 11, 7, 11, 11, 4, 10, 17, 14, 18, 10, 21, 13, 2, 13, 19, 5, 8, 4, 7, 6, 10], such that the sum of the chosen coins adds up to 215. Each coin in the list is unique and can only be used once. Also coins carry a tax value. The tax values for each coin is {17: 9, 107: 7, 13: 11, 10: 7, 19: 12, 9: 2, 4: 4, 14: 8, 21: 14, 6: 2, 5: 3, 2: 2, 3: 3, 7: 6, 18: 6, 8: 4, 11: 11}, where the tax for coins of the same value is the same. Also, if the coin chosen is smaller than the previous one, it must have an even value, otherwise, if the coin is larger than or equal to the previous coin chosen, it must have an odd value. The objective is to determine which subset of coins should be selected to minimize the total tax paid. The solution should be presented as a list of numbers, representing the value of the coins chosen in order, with the first coins chosen being in index 0, formatted in Python syntax.
coin_exchange
subset_sum
18
[18, 6, 9, 8, 2, 5, 5, 17, 14, 8, 107, 14, 2]
60
0.03795170783996582
13
31
31
[[14, 19, 3, 8, 5, 107, 4, 10, 9, 2, 11, 7, 11, 11, 4, 10, 17, 14, 18, 10, 21, 13, 2, 13, 19, 5, 8, 4, 7, 6, 10]]
[[14, 19, 3, 8, 5, 107, 4, 10, 9, 2, 11, 7, 11, 11, 4, 10, 17, 14, 18, 10, 21, 13, 2, 13, 19, 5, 8, 4, 7, 6, 10], {"17": 9, "107": 7, "13": 11, "10": 7, "19": 12, "9": 2, "4": 4, "14": 8, "21": 14, "6": 2, "5": 3, "2": 2, "3": 3, "7": 6, "18": 6, "8": 4, "11": 11}, 215]
["[14, 19, 3, 8, 5, 107, 4, 10, 9, 2, 11, 7, 11, 11, 4, 10, 17, 14, 18, 10, 21, 13, 2, 13, 19, 5, 8, 4, 7, 6, 10]", "{17: 9, 107: 7, 13: 11, 10: 7, 19: 12, 9: 2, 4: 4, 14: 8, 21: 14, 6: 2, 5: 3, 2: 2, 3: 3, 7: 6, 18: 6, 8: 4, 11: 11}", "215"]
4
The game of 'Sort It' begins with 3 tubes, each filled with 4 balls of different colors. The goal is to sort the balls by color, with each tube containing balls of only one color. Only one ball can be moved at a time, taken from the top of one tube and placed on top of another. The capacity of each tube (maximum number of balls we can fit in each tube) is 6 balls. It is not allowed to place a ball in a tube that already has 6 balls. The solution should be a list of tuples, each containing, first, the index of the tube from which a ball is taken and, second, the index of the tube to which it is moved, indexing from 0. Given the initial state of the tubes, represented by the lists below (with the leftmost item being the color of the topmost ball in each tube), what is the shortest list of move tuples that will result in all the balls being correctly sorted? [['Red', 'Red', 'Blue', 'Blue'], ['Red', 'Blue', 'Red', 'Green'], ['Green', 'Blue', 'Green', 'Green']]
color_sorting
sorting
7
[[1, 0], [1, 2], [1, 0], [1, 2], [0, 1], [0, 1], [0, 1], [0, 1], [2, 1], [2, 0], [2, 1], [2, 0], [1, 2], [1, 2]]
14
0.22463750839233398
14
6
12
[[["Red", "Red", "Blue", "Blue"], ["Red", "Blue", "Red", "Green"], ["Green", "Blue", "Green", "Green"]], 6]
[[["Red", "Red", "Blue", "Blue"], ["Red", "Blue", "Red", "Green"], ["Green", "Blue", "Green", "Green"]], 6]
["[['Red', 'Red', 'Blue', 'Blue'], ['Red', 'Blue', 'Red', 'Green'], ['Green', 'Blue', 'Green', 'Green']]", "6"]
4
We have a 3x3 numerical grid, with numbers ranging from 20 to 53 (20 included in the range but 53 is not included). The numbers in each row and column must be strictly increasing or decreasing. This means that either first > second > third or first < second < third in each row and column. If a grid cell is marked with an 'x', the number in that position is hidden. The objective is to replace the 'x's with unique integers from the given range, ensuring that each number only appears once in the grid. The replacements must maintain the consecutive order in each row and column. Additionally, the sum of the numbers in the topmost row plus the numbers in the rightmost column plus the numbers in the diagonal connecting the top-left corner of the grid to its bottom-right corner should be minimized. The solution should be given as a list of tuples in Python syntax. Each tuple should represent the replacement of a number with an 'x' number and contain three elements: the row index of the 'x', the column index of the 'x' (both starting from 0), and the value of the number that replaces the 'x'. The initial state of the grid is as follows: Grid: [['x' '32' '37'] ['29' 'x' 'x'] ['x' '46' 'x']]
consecutive_grid
underdetermined_system
8
[[0, 0, 20], [1, 1, 33], [1, 2, 34], [2, 0, 47], [2, 2, 21]]
255
0.18683171272277832
5
33
9
["[['', '32', '37'], ['29', '', ''], ['', '46', '']]", 20, 53]
["[['', '32', '37'], ['29', '', ''], ['', '46', '']]", 20, 53]
["[['', '32', '37'], ['29', '', ''], ['', '46', '']]", "20", "53"]
4
In the magic square problem, a 3x3 grid is filled with unique integers ranging from 20 to 39. Some numbers are already given, while others are unknown and represented as 'x'. Sum of column 1 (counting from 0) must be 72, and sum of row 1 must be 95. Also, the sum of the numbers in the diagonal from the top right to the bottom left corner of the grid should equal 71. The goal is to find unique integers (ie each number can be in the final grid only once) in the given range to replace with ‘x’s in the grid below such that the sum of the specified rows, columns, and diagonal equals the given amounts and the sum of all of the numbers in the grid is as low as possible. The solution should be provided as a list of tuples in Python syntax. Each tuple should contain three numbers for each 'x' position: the row index, the column index (both starting from 0), and the value of the unique integer replaced with 'x' at that position. Grid: [['21' 'x' 'x'] ['x' 'x' 'x'] ['x' 'x' 'x']]
magic_square
underdetermined_system
6
[[0, 1, 22], [0, 2, 20], [1, 0, 30], [1, 1, 27], [1, 2, 38], [2, 0, 24], [2, 1, 23], [2, 2, 25]]
230
0.5546069145202637
8
19
9
["[['21', '', ''], ['', '', ''], ['', '', '']]", 3, 20, 39]
["[['21', '', ''], ['', '', ''], ['', '', '']]", 20, 39, [1, 2], [1, 2], [72], [95], 71]
["[['21', '', ''], ['', '', ''], ['', '', '']]", "20", "39", "[None, 72, None]", "[None, 95, None]", "71"]
4
In 'Restricted Sorting', there are 6 stacks each with a capacity of 4 blocks, with 3 stacks filled with blocks of varying shades and the remaining are empty. The challenge is to sort the blocks by shade such that if a stack is not empty, it is stacked with 3 blocks of a single shade. The player can only transfer one block at a time from the top of a stack to an empty stack or to a stack that has only blocks of that shade, without exceeding the stacks’ capacity. Transferring blocks to certain stacks is more expensive than others. The cost of moving one block to the top of each stack is: {0: 2, 1: 6, 2: 5, 3: 2, 4: 5, 5: 7}, where the keys are the index of each stack, indexing from 0. The cost of moving a block is always at least 1. The solution should be a list of tuples, each containing, first, the index of the stack from which a block is picked up from and, second, the index of the stack to which it is transferred, indexing from 0. Given the initial state of the stacks, represented by the lists below (with the leftmost item being the shade of the topmost block in each stack)(and the first stack being the stack at index 0), what is the list of transfer pairs (reported in python syntax) with the least possible cost, that will result in all the blocks being correctly sorted? [['Red', 'Green', 'Yellow', 'Blue'], [], ['Yellow', 'Green', 'Blue', 'Red'], [], ['Green', 'Red', 'Blue', 'Yellow'], []]
restricted_sorting
sorting
1
[[0, 3], [0, 1], [4, 1], [4, 3], [0, 5], [4, 0], [2, 4], [2, 1], [2, 0], [5, 4], [2, 3]]
45
0.024616003036499023
11
30
12
[[["Red", "Green", "Yellow", "Blue"], [], ["Yellow", "Green", "Blue", "Red"], [], ["Green", "Red", "Blue", "Yellow"], []], 4, {"0": 2, "1": 6, "2": 5, "3": 2, "4": 5, "5": 7}]
[[["Red", "Green", "Yellow", "Blue"], [], ["Yellow", "Green", "Blue", "Red"], [], ["Green", "Red", "Blue", "Yellow"], []], 4, {"0": 2, "1": 6, "2": 5, "3": 2, "4": 5, "5": 7}, 3]
["[['Red', 'Green', 'Yellow', 'Blue'], [], ['Yellow', 'Green', 'Blue', 'Red'], [], ['Green', 'Red', 'Blue', 'Yellow'], []]", "{0: 2, 1: 6, 2: 5, 3: 2, 4: 5, 5: 7}", "4", "3"]
4
Using the provided matrix map of a city, where numbers represent travel time in minutes (all numbers are positive integers) and 'x' marks closed workshops, find the quickest route for Ben to travel from his current workshop at index (2, 0) to his destination workshop at index (5, 6), indexing from 0. Ben's car can move north, south, east, or west from a given crossroad, provided there's no x in that direction. Also, there are 3 districts in the city with district 1 covering rows 0 to 2, district 2 covering rows 3 to 4, and district 3 covering rows 5 to 8. Ben has to visit at least 1 workshop in each district on his path to the destination. The roads are bidirectional. The answer should be a list of tuples (in Python syntax) indicating the index of workshops on Ben's path. The start and end workshops must be included in the path. [18 9 2 x 9 14 x 1 x] [3 14 18 7 x 3 x 2 19] [6 18 20 3 13 x 6 10 x] [20 x 12 4 14 6 x x x] [7 18 8 5 19 3 x x 7] [15 9 14 16 5 14 6 x x] [x 5 13 x x 19 x x 2] [x x 8 x x x x 17 12] [x 1 x x 3 19 x 2 18]
traffic
pathfinding
1
[[2, 0], [2, 1], [2, 2], [2, 3], [3, 3], [3, 4], [3, 5], [4, 5], [5, 5], [5, 6]]
88
0.02671647071838379
10
4
4
[[["18", "9", "2", "x", "9", "14", "x", "1", "x"], ["3", "14", "18", "7", "x", "3", "x", "2", "19"], ["6", "18", "20", "3", "13", "x", "6", "10", "x"], ["20", "x", "12", "4", "14", "6", "x", "x", "x"], ["7", "18", "8", "5", "19", "3", "x", "x", "7"], ["15", "9", "14", "16", "5", "14", "6", "x", "x"], ["x", "5", "13", "x", "x", "19", "x", "x", "2"], ["x", "x", "8", "x", "x", "x", "x", "17", "12"], ["x", "1", "x", "x", "3", "19", "x", "2", "18"]]]
[[["18", "9", "2", "x", "9", "14", "x", "1", "x"], ["3", "14", "18", "7", "x", "3", "x", "2", "19"], ["6", "18", "20", "3", "13", "x", "6", "10", "x"], ["20", "x", "12", "4", "14", "6", "x", "x", "x"], ["7", "18", "8", "5", "19", "3", "x", "x", "7"], ["15", "9", "14", "16", "5", "14", "6", "x", "x"], ["x", "5", "13", "x", "x", "19", "x", "x", "2"], ["x", "x", "8", "x", "x", "x", "x", "17", "12"], ["x", "1", "x", "x", "3", "19", "x", "2", "18"]], [2, 0], [5, 6], 2, 4]
["[['18', '9', '2', 'x', '9', '14', 'x', '1', 'x'], ['3', '14', '18', '7', 'x', '3', 'x', '2', '19'], ['6', '18', '20', '3', '13', 'x', '6', '10', 'x'], ['20', 'x', '12', '4', '14', '6', 'x', 'x', 'x'], ['7', '18', '8', '5', '19', '3', 'x', 'x', '7'], ['15', '9', '14', '16', '5', '14', '6', 'x', 'x'], ['x', '5', '13', 'x', 'x', '19', 'x', 'x', '2'], ['x', 'x', '8', 'x', 'x', 'x', 'x', '17', '12'], ['x', '1', 'x', 'x', '3', '19', 'x', '2', '18']]", "(2, 0)", "(5, 6)", "2", "4"]
4
Alex is at a trampoline park with a grid of mini trampolines, arranged in a square of 9x9. Some trampolines are broken and unusable. A map of the park is provided below, with 1 indicating a broken trampoline and 0 indicating a functional one. Alex can jump to any of the eight adjacent trampolines, as long as they are not broken. However, Alex must make excatly 3 diagonal jumps, no more, no less, on his path to his destination. He is currently on the trampoline at position (8, 1) (positions are counted from 0, left to right, top to bottom) and wants to reach the trampoline at position (0, 5). What is the shortest sequence of trampolines he should jump on to reach his destination (including the first and final trampolines)? The answer should be a list of tuples, in Python syntax, indicating the row and column of each trampoline Alex jumps on. 0 0 1 0 0 0 0 0 1 0 1 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 0 1 0 1 0 1 1 1 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1 1 0 1 0 0 1 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1
trampoline_matrix
pathfinding
9
[[8, 1], [7, 1], [6, 2], [5, 3], [5, 4], [4, 5], [3, 5], [3, 6], [3, 7], [3, 8], [2, 8], [1, 8], [1, 7], [1, 6], [0, 6], [0, 5]]
16
0.026789426803588867
16
8
2
["[[0, 0, 1, 0, 0, 0, 0, 0, 1], [0, 1, 0, 1, 1, 0, 0, 0, 0], [1, 1, 1, 1, 1, 1, 1, 1, 0], [1, 1, 0, 1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 1, 0, 1, 1, 1], [1, 0, 1, 0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 1, 1, 0, 1], [0, 0, 1, 0, 1, 1, 1, 1, 1], [1, 0, 0, 1, 1, 0, 0, 1, 1]]", 3]
["[[0, 0, 1, 0, 0, 0, 0, 0, 1], [0, 1, 0, 1, 1, 0, 0, 0, 0], [1, 1, 1, 1, 1, 1, 1, 1, 0], [1, 1, 0, 1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 1, 0, 1, 1, 1], [1, 0, 1, 0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 1, 1, 0, 1], [0, 0, 1, 0, 1, 1, 1, 1, 1], [1, 0, 0, 1, 1, 0, 0, 1, 1]]", [8, 1], [0, 5], 3]
["[[0, 0, 1, 0, 0, 0, 0, 0, 1], [0, 1, 0, 1, 1, 0, 0, 0, 0], [1, 1, 1, 1, 1, 1, 1, 1, 0], [1, 1, 0, 1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 1, 0, 1, 1, 1], [1, 0, 1, 0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 1, 1, 0, 1], [0, 0, 1, 0, 1, 1, 1, 1, 1], [1, 0, 0, 1, 1, 0, 0, 1, 1]]", "(8, 1)", "(0, 5)", "3"]
4
Given 5 labeled water jugs with capacities 36, 88, 132, 70, 87, 111 liters, we aim to fill 3 unlabeled buckets, numbered 1 to 3 and arranged in a line in ascending order, with 225, 237, 243 liters of water respectively. The amount of water in each unlabeled bucket can not at any point in time exceed the amount of water in the bucket placed after it. Jugs can only be filled to the top and emptied completely, and the unlabeled buckets cannot be overfilled. An action, represented as a tuple ('+', X, Y) or ('-', X, Y), involves adding to or removing water from the unlabeled bucket numbered Y, using the jug with capacity X. Determine the shortest sequence of actions needed to fill the buckets as specified, and present the solution as a list of action tuples in Python syntax.
water_jug
subset_sum
4
[["+", 111, 3], ["+", 132, 3], ["+", 87, 2], ["+", 132, 2], ["-", 70, 2], ["+", 88, 2], ["+", 70, 1], ["+", 132, 1], ["-", 88, 1], ["+", 111, 1]]
10
0.033365726470947266
10
36
3
[[36, 88, 132, 70, 87, 111], [225, 237, 243]]
[[36, 88, 132, 70, 87, 111], [225, 237, 243]]
["[36, 88, 132, 70, 87, 111]", "[225, 237, 243]"]
5
In the 8-puzzle game, you are given a grid with numbered square tiles arranged randomly and one tile missing. The goal is to arrange the tiles in descending order by sliding them into the empty space. The tiles can move in 4 directions: left, right, up, and down. Given the initial state of the puzzle below, where the empty spot is represented as “_”, provide the shortest list of tiles that need to be swapped with the empty spot to achieve the goal state. The goal state is when all tiles are in descending order, with the largest number in the top left corner, and the empty spot is in the bottom right corner. The solution should be a list of numbers in Python format, where each number represents the number on the tile that the empty spot is swapped with at each turn. Initial state of the puzzle: [[78, 98, 89], [38, '_', 7], [58, 95, 70]]
8_puzzle
puzzle
2
[95, 58, 38, 78, 98, 95, 7, 70, 58, 7, 70, 58]
12
0.029888629913330078
12
4
9
[[[78, 98, 89], [38, "_", 7], [58, 95, 70]]]
[[[78, 98, 89], [38, "_", 7], [58, 95, 70]]]
["[[78, 98, 89], [38, '_', 7], [58, 95, 70]]"]
5
In the game 'Sort the Chars', we are given a table of n by m dimensions. This table contains n words, each with m characters, except for the first word which has m - 1 characters. Each character is written on a separate tile. The objective of the game is to rearrange the characters such that row i spells the i-th word in the list, with the blank tile ('_') placed in the top left corner of the board in the end. We can rearrange the tiles by swapping the blank space with any of its 4 diagonal neighboring tiles. Given the list of words and initial state of the board below, where the black space is represented as '_', what is the shortest list of swap actions (reported in python syntax) that can sort the board into the given list of target words? The list must only include the 4 diagonal swap directions: up-right, down-right, up-left, or down-left, representing the direction in ehich the blank space was swpped in. Target words: sax, tass, turp, cade The initial board: [['a', 's', 't', 'x'], ['t', 'r', 's', 'a'], ['s', 'u', '_', 'p'], ['c', 'a', 'd', 'e']]
8_puzzle_words
puzzle
1
["up-left", "down-left", "down-right", "up-right", "up-left", "up-right", "down-right", "down-left", "down-left", "up-left", "up-right", "up-left"]
12
0.1913444995880127
12
4
16
[[["a", "s", "t", "x"], ["t", "r", "s", "a"], ["s", "u", "_", "p"], ["c", "a", "d", "e"]]]
[[["a", "s", "t", "x"], ["t", "r", "s", "a"], ["s", "u", "_", "p"], ["c", "a", "d", "e"]], ["sax", "tass", "turp", "cade"]]
["[['a', 's', 't', 'x'], ['t', 'r', 's', 'a'], ['s', 'u', '_', 'p'], ['c', 'a', 'd', 'e']]", "['sax', 'tass', 'turp', 'cade']"]
5
We have a map of cities, each represented by a letter, and they are connected by one-way roads. The adjacency matrix below shows the connections between the cities. Each row and column represents a city, and a '1' signifies a direct road from the city of the row to the city of the column. The travel time between any two directly connected cities is the same. Currently, we are located in city 'H'. Our task is to visit city Z and city K excatly twice. Determine the quickest route that allows us to visit both these destination cities, ensuring that we stop at the two destinations twice on our path. The sequence in which we visit the destination cities is not important. However, apart from K and Z, we can only visit each city once on our path. Provide the solution as a list of the city names on our path, including the start, in Python syntax. K B Z S X H O E U V K 0 0 1 1 0 0 1 1 0 1 B 0 0 0 0 0 0 0 1 0 1 Z 0 1 0 1 0 1 1 0 0 1 S 0 1 1 0 1 0 0 0 0 0 X 1 1 0 0 0 0 0 0 0 1 H 0 0 0 1 0 0 1 0 1 0 O 0 0 1 1 1 0 0 0 0 0 E 1 0 0 0 0 0 1 0 0 1 U 0 0 0 0 0 0 1 0 0 0 V 0 0 0 0 1 0 0 0 0 0
city_directed_graph
pathfinding
10
["H", "O", "X", "K", "E", "K", "Z", "S", "Z"]
9
0.03679943084716797
9
10
13
[[[0, 0, 1, 1, 0, 0, 1, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1, 0, 1], [0, 1, 0, 1, 0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 1, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 1, 0, 0, 1, 0, 1, 0], [0, 0, 1, 1, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0]], ["K", "B", "Z", "S", "X", "H", "O", "E", "U", "V"], "Z", "K"]
[[[0, 0, 1, 1, 0, 0, 1, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1, 0, 1], [0, 1, 0, 1, 0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 1, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 1, 0, 0, 1, 0, 1, 0], [0, 0, 1, 1, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0]], ["K", "B", "Z", "S", "X", "H", "O", "E", "U", "V"], "H", "Z", "K"]
["[[0, 0, 1, 1, 0, 0, 1, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1, 0, 1], [0, 1, 0, 1, 0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 1, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 1, 0, 0, 1, 0, 1, 0], [0, 0, 1, 1, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0]]", "['K', 'B', 'Z', 'S', 'X', 'H', 'O', 'E', 'U', 'V']", "['H']", "['Z', 'K']"]
5
In the 'taxed coin exchange' problem, you are required to choose a subset of coins from this list [20, 10, 16, 21, 15, 21, 13, 6, 19, 9, 7, 5, 2, 6, 9, 22, 4, 13, 3, 16, 22, 18, 6, 5, 14, 7, 7, 2, 17, 9, 8, 83, 14], such that the sum of the chosen coins adds up to 239. Each coin in the list is unique and can only be used once. Also coins carry a tax value. The tax values for each coin is {22: 18, 21: 1, 14: 12, 10: 8, 8: 6, 17: 9, 6: 4, 7: 6, 9: 3, 20: 13, 19: 12, 5: 3, 13: 8, 4: 3, 83: 14, 3: 2, 18: 1, 2: 1, 16: 2, 15: 3}, where the tax for coins of the same value is the same. Also, if the coin chosen is smaller than the previous one, it must have an even value, otherwise, if the coin is larger than or equal to the previous coin chosen, it must have an odd value. The objective is to determine which subset of coins should be selected to minimize the total tax paid. The solution should be presented as a list of numbers, representing the value of the coins chosen in order, with the first coins chosen being in index 0, formatted in Python syntax.
coin_exchange
subset_sum
19
[18, 21, 16, 2, 15, 2, 9, 9, 21, 83, 16, 6, 4, 17]
48
0.03824257850646973
14
33
33
[[20, 10, 16, 21, 15, 21, 13, 6, 19, 9, 7, 5, 2, 6, 9, 22, 4, 13, 3, 16, 22, 18, 6, 5, 14, 7, 7, 2, 17, 9, 8, 83, 14]]
[[20, 10, 16, 21, 15, 21, 13, 6, 19, 9, 7, 5, 2, 6, 9, 22, 4, 13, 3, 16, 22, 18, 6, 5, 14, 7, 7, 2, 17, 9, 8, 83, 14], {"22": 18, "21": 1, "14": 12, "10": 8, "8": 6, "17": 9, "6": 4, "7": 6, "9": 3, "20": 13, "19": 12, "5": 3, "13": 8, "4": 3, "83": 14, "3": 2, "18": 1, "2": 1, "16": 2, "15": 3}, 239]
["[20, 10, 16, 21, 15, 21, 13, 6, 19, 9, 7, 5, 2, 6, 9, 22, 4, 13, 3, 16, 22, 18, 6, 5, 14, 7, 7, 2, 17, 9, 8, 83, 14]", "{22: 18, 21: 1, 14: 12, 10: 8, 8: 6, 17: 9, 6: 4, 7: 6, 9: 3, 20: 13, 19: 12, 5: 3, 13: 8, 4: 3, 83: 14, 3: 2, 18: 1, 2: 1, 16: 2, 15: 3}", "239"]
5
The game of 'Sort It' begins with 3 tubes, each filled with 4 balls of different colors. The goal is to sort the balls by color, with each tube containing balls of only one color. Only one ball can be moved at a time, taken from the top of one tube and placed on top of another. The capacity of each tube (maximum number of balls we can fit in each tube) is 6 balls. It is not allowed to place a ball in a tube that already has 6 balls. The solution should be a list of tuples, each containing, first, the index of the tube from which a ball is taken and, second, the index of the tube to which it is moved, indexing from 0. Given the initial state of the tubes, represented by the lists below (with the leftmost item being the color of the topmost ball in each tube), what is the shortest list of move tuples that will result in all the balls being correctly sorted? [['Green', 'Blue', 'Blue', 'Blue'], ['Red', 'Green', 'Red', 'Red'], ['Green', 'Red', 'Green', 'Blue']]
color_sorting
sorting
7
[[1, 2], [1, 0], [2, 1], [2, 0], [2, 1], [2, 1], [2, 1], [0, 2], [0, 2], [0, 2], [1, 0], [1, 2]]
12
0.08999156951904297
12
6
12
[[["Green", "Blue", "Blue", "Blue"], ["Red", "Green", "Red", "Red"], ["Green", "Red", "Green", "Blue"]], 6]
[[["Green", "Blue", "Blue", "Blue"], ["Red", "Green", "Red", "Red"], ["Green", "Red", "Green", "Blue"]], 6]
["[['Green', 'Blue', 'Blue', 'Blue'], ['Red', 'Green', 'Red', 'Red'], ['Green', 'Red', 'Green', 'Blue']]", "6"]
5
We have a 3x3 numerical grid, with numbers ranging from 44 to 77 (44 included in the range but 77 is not included). The numbers in each row and column must be strictly increasing or decreasing. This means that either first > second > third or first < second < third in each row and column. If a grid cell is marked with an 'x', the number in that position is hidden. The objective is to replace the 'x's with unique integers from the given range, ensuring that each number only appears once in the grid. The replacements must maintain the consecutive order in each row and column. Additionally, the sum of the numbers in the topmost row plus the numbers in the rightmost column plus the numbers in the diagonal connecting the top-left corner of the grid to its bottom-right corner should be minimized. The solution should be given as a list of tuples in Python syntax. Each tuple should represent the replacement of a number with an 'x' number and contain three elements: the row index of the 'x', the column index of the 'x' (both starting from 0), and the value of the number that replaces the 'x'. The initial state of the grid is as follows: Grid: [['66' 'x' '74'] ['x' '60' '62'] ['x' 'x' 'x']]
consecutive_grid
underdetermined_system
8
[[0, 1, 67], [1, 0, 47], [2, 0, 46], [2, 1, 45], [2, 2, 44]]
557
0.45664143562316895
5
33
9
["[['66', '', '74'], ['', '60', '62'], ['', '', '']]", 44, 77]
["[['66', '', '74'], ['', '60', '62'], ['', '', '']]", 44, 77]
["[['66', '', '74'], ['', '60', '62'], ['', '', '']]", "44", "77"]
5
In the magic square problem, a 3x3 grid is filled with unique integers ranging from 20 to 39. Some numbers are already given, while others are unknown and represented as 'x'. Sum of column 1 (counting from 0) must be 82, and sum of row 1 must be 82. Also, the sum of the numbers in the diagonal from the top right to the bottom left corner of the grid should equal 82. The goal is to find unique integers (ie each number can be in the final grid only once) in the given range to replace with ‘x’s in the grid below such that the sum of the specified rows, columns, and diagonal equals the given amounts and the sum of all of the numbers in the grid is as low as possible. The solution should be provided as a list of tuples in Python syntax. Each tuple should contain three numbers for each 'x' position: the row index, the column index (both starting from 0), and the value of the unique integer replaced with 'x' at that position. Grid: [['x' 'x' 'x'] ['x' 'x' 'x'] ['22' 'x' 'x']]
magic_square
underdetermined_system
7
[[0, 0, 20], [0, 1, 23], [0, 2, 27], [1, 0, 24], [1, 1, 33], [1, 2, 25], [2, 1, 26], [2, 2, 21]]
221
0.6350831985473633
8
19
9
["[['', '', ''], ['', '', ''], ['22', '', '']]", 3, 20, 39]
["[['', '', ''], ['', '', ''], ['22', '', '']]", 20, 39, [1, 2], [1, 2], [82], [82], 82]
["[['', '', ''], ['', '', ''], ['22', '', '']]", "20", "39", "[None, 82, None]", "[None, 82, None]", "82"]
5
In 'Restricted Sorting', there are 6 stacks each with a capacity of 4 blocks, with 3 stacks filled with blocks of varying shades and the remaining are empty. The challenge is to sort the blocks by shade such that if a stack is not empty, it is stacked with 3 blocks of a single shade. The player can only transfer one block at a time from the top of a stack to an empty stack or to a stack that has only blocks of that shade, without exceeding the stacks’ capacity. Transferring blocks to certain stacks is more expensive than others. The cost of moving one block to the top of each stack is: {0: 4, 1: 1, 2: 7, 3: 3, 4: 2, 5: 2}, where the keys are the index of each stack, indexing from 0. The cost of moving a block is always at least 1. The solution should be a list of tuples, each containing, first, the index of the stack from which a block is picked up from and, second, the index of the stack to which it is transferred, indexing from 0. Given the initial state of the stacks, represented by the lists below (with the leftmost item being the shade of the topmost block in each stack)(and the first stack being the stack at index 0), what is the list of transfer pairs (reported in python syntax) with the least possible cost, that will result in all the blocks being correctly sorted? [['Red', 'Green', 'Green', 'Yellow'], [], ['Blue', 'Blue', 'Green', 'Red'], [], ['Blue', 'Yellow', 'Red', 'Yellow'], []]
restricted_sorting
sorting
1
[[4, 3], [0, 1], [0, 5], [0, 5], [4, 0], [4, 1], [4, 0], [2, 3], [2, 3], [2, 5], [2, 1]]
26
0.1328294277191162
11
30
12
[[["Red", "Green", "Green", "Yellow"], [], ["Blue", "Blue", "Green", "Red"], [], ["Blue", "Yellow", "Red", "Yellow"], []], 4, {"0": 4, "1": 1, "2": 7, "3": 3, "4": 2, "5": 2}]
[[["Red", "Green", "Green", "Yellow"], [], ["Blue", "Blue", "Green", "Red"], [], ["Blue", "Yellow", "Red", "Yellow"], []], 4, {"0": 4, "1": 1, "2": 7, "3": 3, "4": 2, "5": 2}, 3]
["[['Red', 'Green', 'Green', 'Yellow'], [], ['Blue', 'Blue', 'Green', 'Red'], [], ['Blue', 'Yellow', 'Red', 'Yellow'], []]", "{0: 4, 1: 1, 2: 7, 3: 3, 4: 2, 5: 2}", "4", "3"]
5
Using the provided matrix map of a city, where numbers represent travel time in minutes (all numbers are positive integers) and 'x' marks closed workshops, find the quickest route for Ben to travel from his current workshop at index (5, 0) to his destination workshop at index (3, 7), indexing from 0. Ben's car can move north, south, east, or west from a given crossroad, provided there's no x in that direction. Also, there are 3 districts in the city with district 1 covering rows 0 to 3, district 2 covering rows 4 to 4, and district 3 covering rows 5 to 8. Ben has to visit at least 1 workshop in each district on his path to the destination. The roads are bidirectional. The answer should be a list of tuples (in Python syntax) indicating the index of workshops on Ben's path. The start and end workshops must be included in the path. [8 x 4 7 10 x 12 5 x] [x x 4 11 16 4 12 15 x] [3 x 16 16 4 x x x x] [11 6 3 12 8 2 19 14 x] [20 18 19 x 4 18 x 15 13] [8 15 x 16 11 x 10 14 1] [x x x 19 x x x x x] [7 x x x x x 1 7 8] [15 3 x x x x x x x]
traffic
pathfinding
1
[[5, 0], [4, 0], [3, 0], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [3, 6], [3, 7]]
95
0.022897005081176758
10
4
4
[[["8", "x", "4", "7", "10", "x", "12", "5", "x"], ["x", "x", "4", "11", "16", "4", "12", "15", "x"], ["3", "x", "16", "16", "4", "x", "x", "x", "x"], ["11", "6", "3", "12", "8", "2", "19", "14", "x"], ["20", "18", "19", "x", "4", "18", "x", "15", "13"], ["8", "15", "x", "16", "11", "x", "10", "14", "1"], ["x", "x", "x", "19", "x", "x", "x", "x", "x"], ["7", "x", "x", "x", "x", "x", "1", "7", "8"], ["15", "3", "x", "x", "x", "x", "x", "x", "x"]]]
[[["8", "x", "4", "7", "10", "x", "12", "5", "x"], ["x", "x", "4", "11", "16", "4", "12", "15", "x"], ["3", "x", "16", "16", "4", "x", "x", "x", "x"], ["11", "6", "3", "12", "8", "2", "19", "14", "x"], ["20", "18", "19", "x", "4", "18", "x", "15", "13"], ["8", "15", "x", "16", "11", "x", "10", "14", "1"], ["x", "x", "x", "19", "x", "x", "x", "x", "x"], ["7", "x", "x", "x", "x", "x", "1", "7", "8"], ["15", "3", "x", "x", "x", "x", "x", "x", "x"]], [5, 0], [3, 7], 3, 4]
["[['8', 'x', '4', '7', '10', 'x', '12', '5', 'x'], ['x', 'x', '4', '11', '16', '4', '12', '15', 'x'], ['3', 'x', '16', '16', '4', 'x', 'x', 'x', 'x'], ['11', '6', '3', '12', '8', '2', '19', '14', 'x'], ['20', '18', '19', 'x', '4', '18', 'x', '15', '13'], ['8', '15', 'x', '16', '11', 'x', '10', '14', '1'], ['x', 'x', 'x', '19', 'x', 'x', 'x', 'x', 'x'], ['7', 'x', 'x', 'x', 'x', 'x', '1', '7', '8'], ['15', '3', 'x', 'x', 'x', 'x', 'x', 'x', 'x']]", "(5, 0)", "(3, 7)", "3", "4"]
5
Alex is at a trampoline park with a grid of mini trampolines, arranged in a square of 9x9. Some trampolines are broken and unusable. A map of the park is provided below, with 1 indicating a broken trampoline and 0 indicating a functional one. Alex can jump to any of the eight adjacent trampolines, as long as they are not broken. However, Alex must make excatly 3 diagonal jumps, no more, no less, on his path to his destination. He is currently on the trampoline at position (0, 0) (positions are counted from 0, left to right, top to bottom) and wants to reach the trampoline at position (5, 7). What is the shortest sequence of trampolines he should jump on to reach his destination (including the first and final trampolines)? The answer should be a list of tuples, in Python syntax, indicating the row and column of each trampoline Alex jumps on. 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 1 0 1 1
trampoline_matrix
pathfinding
9
[[0, 0], [0, 1], [0, 2], [0, 3], [1, 3], [1, 4], [2, 5], [3, 5], [4, 6], [5, 7]]
10
0.031831979751586914
10
8
2
["[[0, 0, 0, 0, 1, 1, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 1, 1, 1, 1, 0, 0, 0, 1], [1, 1, 1, 1, 1, 0, 0, 1, 0], [1, 1, 0, 1, 1, 0, 0, 0, 0], [0, 1, 1, 0, 1, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0, 1, 1], [0, 0, 0, 1, 1, 0, 0, 1, 1], [1, 0, 1, 0, 1, 1, 0, 1, 1]]", 3]
["[[0, 0, 0, 0, 1, 1, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 1, 1, 1, 1, 0, 0, 0, 1], [1, 1, 1, 1, 1, 0, 0, 1, 0], [1, 1, 0, 1, 1, 0, 0, 0, 0], [0, 1, 1, 0, 1, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0, 1, 1], [0, 0, 0, 1, 1, 0, 0, 1, 1], [1, 0, 1, 0, 1, 1, 0, 1, 1]]", [0, 0], [5, 7], 3]
["[[0, 0, 0, 0, 1, 1, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 1, 1, 1, 1, 0, 0, 0, 1], [1, 1, 1, 1, 1, 0, 0, 1, 0], [1, 1, 0, 1, 1, 0, 0, 0, 0], [0, 1, 1, 0, 1, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0, 1, 1], [0, 0, 0, 1, 1, 0, 0, 1, 1], [1, 0, 1, 0, 1, 1, 0, 1, 1]]", "(0, 0)", "(5, 7)", "3"]
5
Given 5 labeled water jugs with capacities 145, 101, 17, 75, 102, 129 liters, we aim to fill 3 unlabeled buckets, numbered 1 to 3 and arranged in a line in ascending order, with 260, 287, 413 liters of water respectively. The amount of water in each unlabeled bucket can not at any point in time exceed the amount of water in the bucket placed after it. Jugs can only be filled to the top and emptied completely, and the unlabeled buckets cannot be overfilled. An action, represented as a tuple ('+', X, Y) or ('-', X, Y), involves adding to or removing water from the unlabeled bucket numbered Y, using the jug with capacity X. Determine the shortest sequence of actions needed to fill the buckets as specified, and present the solution as a list of action tuples in Python syntax.
water_jug
subset_sum
4
[["+", 17, 3], ["+", 101, 3], ["+", 145, 3], ["+", 75, 3], ["+", 75, 3], ["+", 101, 2], ["+", 101, 2], ["-", 17, 2], ["+", 102, 2], ["+", 101, 1], ["-", 17, 1], ["+", 75, 1], ["+", 101, 1]]
13
0.04054903984069824
13
36
3
[[145, 101, 17, 75, 102, 129], [260, 287, 413]]
[[145, 101, 17, 75, 102, 129], [260, 287, 413]]
["[145, 101, 17, 75, 102, 129]", "[260, 287, 413]"]
6
In the 8-puzzle game, you are given a grid with numbered square tiles arranged randomly and one tile missing. The goal is to arrange the tiles in descending order by sliding them into the empty space. The tiles can move in 4 directions: left, right, up, and down. Given the initial state of the puzzle below, where the empty spot is represented as “_”, provide the shortest list of tiles that need to be swapped with the empty spot to achieve the goal state. The goal state is when all tiles are in descending order, with the largest number in the top left corner, and the empty spot is in the bottom right corner. The solution should be a list of numbers in Python format, where each number represents the number on the tile that the empty spot is swapped with at each turn. Initial state of the puzzle: [[46, 82, 90], [36, 41, 38], [28, 55, '_']]
8_puzzle
puzzle
2
[38, 41, 82, 90, 41, 82, 55, 28, 36, 46, 90, 41, 82, 55, 41, 82, 55, 38]
18
0.046418190002441406
18
4
9
[[[46, 82, 90], [36, 41, 38], [28, 55, "_"]]]
[[[46, 82, 90], [36, 41, 38], [28, 55, "_"]]]
["[[46, 82, 90], [36, 41, 38], [28, 55, '_']]"]
6
In the game 'Sort the Chars', we are given a table of n by m dimensions. This table contains n words, each with m characters, except for the first word which has m - 1 characters. Each character is written on a separate tile. The objective of the game is to rearrange the characters such that row i spells the i-th word in the list, with the blank tile ('_') placed in the top left corner of the board in the end. We can rearrange the tiles by swapping the blank space with any of its 4 diagonal neighboring tiles. Given the list of words and initial state of the board below, where the black space is represented as '_', what is the shortest list of swap actions (reported in python syntax) that can sort the board into the given list of target words? The list must only include the 4 diagonal swap directions: up-right, down-right, up-left, or down-left, representing the direction in ehich the blank space was swpped in. Target words: cho, slod, syce, halt The initial board: [['l', 'c', '_', 'o'], ['s', 'c', 'o', 'h'], ['d', 'y', 'a', 'e'], ['h', 's', 'l', 't']]
8_puzzle_words
puzzle
1
["down-left", "down-left", "down-right", "up-right", "up-left", "up-right", "down-right", "down-left", "up-left", "up-left"]
10
0.16235899925231934
10
4
16
[[["l", "c", "_", "o"], ["s", "c", "o", "h"], ["d", "y", "a", "e"], ["h", "s", "l", "t"]]]
[[["l", "c", "_", "o"], ["s", "c", "o", "h"], ["d", "y", "a", "e"], ["h", "s", "l", "t"]], ["cho", "slod", "syce", "halt"]]
["[['l', 'c', '_', 'o'], ['s', 'c', 'o', 'h'], ['d', 'y', 'a', 'e'], ['h', 's', 'l', 't']]", "['cho', 'slod', 'syce', 'halt']"]
6
We have a map of cities, each represented by a letter, and they are connected by one-way roads. The adjacency matrix below shows the connections between the cities. Each row and column represents a city, and a '1' signifies a direct road from the city of the row to the city of the column. The travel time between any two directly connected cities is the same. Currently, we are located in city 'N'. Our task is to visit city D and city F excatly twice. Determine the quickest route that allows us to visit both these destination cities, ensuring that we stop at the two destinations twice on our path. The sequence in which we visit the destination cities is not important. However, apart from F and D, we can only visit each city once on our path. Provide the solution as a list of the city names on our path, including the start, in Python syntax. F I O J K D N R M B F 0 0 0 1 1 0 0 0 1 0 I 1 0 0 0 0 0 0 0 0 1 O 0 0 0 1 0 1 0 0 1 1 J 1 0 1 0 0 0 0 0 0 0 K 0 0 1 0 0 1 0 0 0 0 D 0 1 1 0 0 0 0 1 0 0 N 0 1 0 0 0 0 0 1 0 0 R 0 0 0 1 0 0 0 0 1 0 M 1 1 0 1 0 0 0 0 0 0 B 1 0 0 1 0 1 0 0 1 0
city_directed_graph
pathfinding
10
["N", "I", "F", "J", "F", "K", "D", "O", "D"]
9
0.028170108795166016
9
10
13
[[[0, 0, 0, 1, 1, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 1, 0, 1, 0, 0, 1, 1], [1, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 1, 0, 0, 0, 0], [0, 1, 1, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 1, 0], [1, 1, 0, 1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 1, 0, 0, 1, 0]], ["F", "I", "O", "J", "K", "D", "N", "R", "M", "B"], "D", "F"]
[[[0, 0, 0, 1, 1, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 1, 0, 1, 0, 0, 1, 1], [1, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 1, 0, 0, 0, 0], [0, 1, 1, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 1, 0], [1, 1, 0, 1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 1, 0, 0, 1, 0]], ["F", "I", "O", "J", "K", "D", "N", "R", "M", "B"], "N", "D", "F"]
["[[0, 0, 0, 1, 1, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 1, 0, 1, 0, 0, 1, 1], [1, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 1, 0, 0, 0, 0], [0, 1, 1, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 1, 0], [1, 1, 0, 1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 1, 0, 0, 1, 0]]", "['F', 'I', 'O', 'J', 'K', 'D', 'N', 'R', 'M', 'B']", "['N']", "['D', 'F']"]
6
In the 'taxed coin exchange' problem, you are required to choose a subset of coins from this list [20, 25, 10, 4, 13, 3, 10, 17, 5, 25, 17, 2, 19, 24, 25, 10, 19, 8, 2, 20, 15, 84, 23, 87, 19, 8, 12, 17, 23, 6], such that the sum of the chosen coins adds up to 250. Each coin in the list is unique and can only be used once. Also coins carry a tax value. The tax values for each coin is {12: 12, 19: 8, 84: 20, 24: 1, 5: 1, 3: 1, 4: 2, 87: 7, 10: 5, 25: 7, 6: 5, 20: 5, 15: 7, 23: 17, 13: 4, 17: 16, 2: 1, 8: 5}, where the tax for coins of the same value is the same. Also, if the coin chosen is smaller than the previous one, it must have an even value, otherwise, if the coin is larger than or equal to the previous coin chosen, it must have an odd value. The objective is to determine which subset of coins should be selected to minimize the total tax paid. The solution should be presented as a list of numbers, representing the value of the coins chosen in order, with the first coins chosen being in index 0, formatted in Python syntax.
coin_exchange
subset_sum
20
[3, 5, 87, 84, 24, 20, 25, 2]
43
0.036344051361083984
8
30
30
[[20, 25, 10, 4, 13, 3, 10, 17, 5, 25, 17, 2, 19, 24, 25, 10, 19, 8, 2, 20, 15, 84, 23, 87, 19, 8, 12, 17, 23, 6]]
[[20, 25, 10, 4, 13, 3, 10, 17, 5, 25, 17, 2, 19, 24, 25, 10, 19, 8, 2, 20, 15, 84, 23, 87, 19, 8, 12, 17, 23, 6], {"12": 12, "19": 8, "84": 20, "24": 1, "5": 1, "3": 1, "4": 2, "87": 7, "10": 5, "25": 7, "6": 5, "20": 5, "15": 7, "23": 17, "13": 4, "17": 16, "2": 1, "8": 5}, 250]
["[20, 25, 10, 4, 13, 3, 10, 17, 5, 25, 17, 2, 19, 24, 25, 10, 19, 8, 2, 20, 15, 84, 23, 87, 19, 8, 12, 17, 23, 6]", "{12: 12, 19: 8, 84: 20, 24: 1, 5: 1, 3: 1, 4: 2, 87: 7, 10: 5, 25: 7, 6: 5, 20: 5, 15: 7, 23: 17, 13: 4, 17: 16, 2: 1, 8: 5}", "250"]
6
The game of 'Sort It' begins with 3 tubes, each filled with 4 balls of different colors. The goal is to sort the balls by color, with each tube containing balls of only one color. Only one ball can be moved at a time, taken from the top of one tube and placed on top of another. The capacity of each tube (maximum number of balls we can fit in each tube) is 6 balls. It is not allowed to place a ball in a tube that already has 6 balls. The solution should be a list of tuples, each containing, first, the index of the tube from which a ball is taken and, second, the index of the tube to which it is moved, indexing from 0. Given the initial state of the tubes, represented by the lists below (with the leftmost item being the color of the topmost ball in each tube), what is the shortest list of move tuples that will result in all the balls being correctly sorted? [['Blue', 'Green', 'Blue', 'Red'], ['Red', 'Blue', 'Green', 'Red'], ['Green', 'Green', 'Blue', 'Red']]
color_sorting
sorting
7
[[0, 1], [0, 2], [0, 1], [0, 2], [1, 0], [1, 0], [1, 0], [1, 0], [2, 0], [1, 2], [1, 0], [2, 1], [2, 1], [2, 1], [2, 1], [2, 1], [0, 2], [0, 2], [0, 1], [0, 2], [1, 0], [1, 0]]
22
7.465789318084717
22
6
12
[[["Blue", "Green", "Blue", "Red"], ["Red", "Blue", "Green", "Red"], ["Green", "Green", "Blue", "Red"]], 6]
[[["Blue", "Green", "Blue", "Red"], ["Red", "Blue", "Green", "Red"], ["Green", "Green", "Blue", "Red"]], 6]
["[['Blue', 'Green', 'Blue', 'Red'], ['Red', 'Blue', 'Green', 'Red'], ['Green', 'Green', 'Blue', 'Red']]", "6"]
6
We have a 3x3 numerical grid, with numbers ranging from 41 to 74 (41 included in the range but 74 is not included). The numbers in each row and column must be strictly increasing or decreasing. This means that either first > second > third or first < second < third in each row and column. If a grid cell is marked with an 'x', the number in that position is hidden. The objective is to replace the 'x's with unique integers from the given range, ensuring that each number only appears once in the grid. The replacements must maintain the consecutive order in each row and column. Additionally, the sum of the numbers in the topmost row plus the numbers in the rightmost column plus the numbers in the diagonal connecting the top-left corner of the grid to its bottom-right corner should be minimized. The solution should be given as a list of tuples in Python syntax. Each tuple should represent the replacement of a number with an 'x' number and contain three elements: the row index of the 'x', the column index of the 'x' (both starting from 0), and the value of the number that replaces the 'x'. The initial state of the grid is as follows: Grid: [['41' '48' 'x'] ['x' 'x' '63'] ['x' 'x' '67']]
consecutive_grid
underdetermined_system
8
[[0, 2, 49], [1, 0, 42], [1, 1, 45], [2, 0, 43], [2, 1, 44]]
470
0.18347549438476562
5
33
9
["[['41', '48', ''], ['', '', '63'], ['', '', '67']]", 41, 74]
["[['41', '48', ''], ['', '', '63'], ['', '', '67']]", 41, 74]
["[['41', '48', ''], ['', '', '63'], ['', '', '67']]", "41", "74"]
6
In the magic square problem, a 3x3 grid is filled with unique integers ranging from 19 to 43. Some numbers are already given, while others are unknown and represented as 'x'. Sum of column 1 (counting from 0) must be 82, and sum of row 1 must be 82. Also, the sum of the numbers in the diagonal from the top right to the bottom left corner of the grid should equal 96. The goal is to find unique integers (ie each number can be in the final grid only once) in the given range to replace with ‘x’s in the grid below such that the sum of the specified rows, columns, and diagonal equals the given amounts and the sum of all of the numbers in the grid is as low as possible. The solution should be provided as a list of tuples in Python syntax. Each tuple should contain three numbers for each 'x' position: the row index, the column index (both starting from 0), and the value of the unique integer replaced with 'x' at that position. Grid: [['x' '20' 'x'] ['x' 'x' 'x'] ['x' 'x' '33']]
magic_square
underdetermined_system
7
[[0, 0, 19], [0, 2, 25], [1, 0, 21], [1, 1, 39], [1, 2, 22], [2, 0, 32], [2, 1, 23]]
234
7.9251344203948975
7
19
9
["[['', '20', ''], ['', '', ''], ['', '', '33']]", 3, 19, 43]
["[['', '20', ''], ['', '', ''], ['', '', '33']]", 19, 43, [1, 2], [1, 2], [82], [82], 96]
["[['', '20', ''], ['', '', ''], ['', '', '33']]", "19", "43", "[None, 82, None]", "[None, 82, None]", "96"]
6
In 'Restricted Sorting', there are 6 stacks each with a capacity of 4 blocks, with 3 stacks filled with blocks of varying shades and the remaining are empty. The challenge is to sort the blocks by shade such that if a stack is not empty, it is stacked with 3 blocks of a single shade. The player can only transfer one block at a time from the top of a stack to an empty stack or to a stack that has only blocks of that shade, without exceeding the stacks’ capacity. Transferring blocks to certain stacks is more expensive than others. The cost of moving one block to the top of each stack is: {0: 7, 1: 3, 2: 5, 3: 5, 4: 1, 5: 3}, where the keys are the index of each stack, indexing from 0. The cost of moving a block is always at least 1. The solution should be a list of tuples, each containing, first, the index of the stack from which a block is picked up from and, second, the index of the stack to which it is transferred, indexing from 0. Given the initial state of the stacks, represented by the lists below (with the leftmost item being the shade of the topmost block in each stack)(and the first stack being the stack at index 0), what is the list of transfer pairs (reported in python syntax) with the least possible cost, that will result in all the blocks being correctly sorted? [['Yellow', 'Blue', 'Blue', 'Green'], [], ['Yellow', 'Green', 'Red', 'Red'], ['Green', 'Red', 'Yellow', 'Blue'], [], []]
restricted_sorting
sorting
1
[[0, 5], [2, 5], [2, 4], [3, 4], [3, 2], [0, 1], [0, 1], [3, 5], [3, 1], [0, 4]]
26
0.06659841537475586
10
30
12
[[["Yellow", "Blue", "Blue", "Green"], [], ["Yellow", "Green", "Red", "Red"], ["Green", "Red", "Yellow", "Blue"], [], []], 4, {"0": 7, "1": 3, "2": 5, "3": 5, "4": 1, "5": 3}]
[[["Yellow", "Blue", "Blue", "Green"], [], ["Yellow", "Green", "Red", "Red"], ["Green", "Red", "Yellow", "Blue"], [], []], 4, {"0": 7, "1": 3, "2": 5, "3": 5, "4": 1, "5": 3}, 3]
["[['Yellow', 'Blue', 'Blue', 'Green'], [], ['Yellow', 'Green', 'Red', 'Red'], ['Green', 'Red', 'Yellow', 'Blue'], [], []]", "{0: 7, 1: 3, 2: 5, 3: 5, 4: 1, 5: 3}", "4", "3"]
6
Using the provided matrix map of a city, where numbers represent travel time in minutes (all numbers are positive integers) and 'x' marks closed workshops, find the quickest route for Ben to travel from his current workshop at index (5, 5) to his destination workshop at index (1, 0), indexing from 0. Ben's car can move north, south, east, or west from a given crossroad, provided there's no x in that direction. Also, there are 3 districts in the city with district 1 covering rows 0 to 1, district 2 covering rows 2 to 4, and district 3 covering rows 5 to 8. Ben has to visit at least 1 workshop in each district on his path to the destination. The roads are bidirectional. The answer should be a list of tuples (in Python syntax) indicating the index of workshops on Ben's path. The start and end workshops must be included in the path. [6 16 x x 15 x 14 11 x] [5 2 1 x 3 x 4 13 x] [x 7 x x 13 x 10 14 5] [6 16 x 13 6 x 16 x x] [x 1 17 x 6 3 x x 3] [11 10 7 19 15 8 12 6 x] [18 x x 9 8 9 x x 13] [4 x x x 18 20 x x x] [x x 8 x 18 2 6 13 x]
traffic
pathfinding
1
[[5, 5], [5, 4], [5, 3], [5, 2], [5, 1], [4, 1], [3, 1], [2, 1], [1, 1], [1, 0]]
82
0.027018070220947266
10
4
4
[[["6", "16", "x", "x", "15", "x", "14", "11", "x"], ["5", "2", "1", "x", "3", "x", "4", "13", "x"], ["x", "7", "x", "x", "13", "x", "10", "14", "5"], ["6", "16", "x", "13", "6", "x", "16", "x", "x"], ["x", "1", "17", "x", "6", "3", "x", "x", "3"], ["11", "10", "7", "19", "15", "8", "12", "6", "x"], ["18", "x", "x", "9", "8", "9", "x", "x", "13"], ["4", "x", "x", "x", "18", "20", "x", "x", "x"], ["x", "x", "8", "x", "18", "2", "6", "13", "x"]]]
[[["6", "16", "x", "x", "15", "x", "14", "11", "x"], ["5", "2", "1", "x", "3", "x", "4", "13", "x"], ["x", "7", "x", "x", "13", "x", "10", "14", "5"], ["6", "16", "x", "13", "6", "x", "16", "x", "x"], ["x", "1", "17", "x", "6", "3", "x", "x", "3"], ["11", "10", "7", "19", "15", "8", "12", "6", "x"], ["18", "x", "x", "9", "8", "9", "x", "x", "13"], ["4", "x", "x", "x", "18", "20", "x", "x", "x"], ["x", "x", "8", "x", "18", "2", "6", "13", "x"]], [5, 5], [1, 0], 1, 4]
["[['6', '16', 'x', 'x', '15', 'x', '14', '11', 'x'], ['5', '2', '1', 'x', '3', 'x', '4', '13', 'x'], ['x', '7', 'x', 'x', '13', 'x', '10', '14', '5'], ['6', '16', 'x', '13', '6', 'x', '16', 'x', 'x'], ['x', '1', '17', 'x', '6', '3', 'x', 'x', '3'], ['11', '10', '7', '19', '15', '8', '12', '6', 'x'], ['18', 'x', 'x', '9', '8', '9', 'x', 'x', '13'], ['4', 'x', 'x', 'x', '18', '20', 'x', 'x', 'x'], ['x', 'x', '8', 'x', '18', '2', '6', '13', 'x']]", "(5, 5)", "(1, 0)", "1", "4"]
6
Alex is at a trampoline park with a grid of mini trampolines, arranged in a square of 9x9. Some trampolines are broken and unusable. A map of the park is provided below, with 1 indicating a broken trampoline and 0 indicating a functional one. Alex can jump to any of the eight adjacent trampolines, as long as they are not broken. However, Alex must make excatly 3 diagonal jumps, no more, no less, on his path to his destination. He is currently on the trampoline at position (0, 6) (positions are counted from 0, left to right, top to bottom) and wants to reach the trampoline at position (8, 2). What is the shortest sequence of trampolines he should jump on to reach his destination (including the first and final trampolines)? The answer should be a list of tuples, in Python syntax, indicating the row and column of each trampoline Alex jumps on. 1 0 1 0 0 1 0 1 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 0 0 1 0 0 0 1 1 1 1
trampoline_matrix
pathfinding
9
[[0, 6], [1, 6], [2, 5], [3, 5], [4, 4], [5, 3], [6, 3], [7, 3], [7, 2], [8, 2]]
10
0.02278614044189453
10
8
2
["[[1, 0, 1, 0, 0, 1, 0, 1, 0], [1, 1, 1, 0, 1, 1, 0, 0, 1], [1, 1, 1, 1, 1, 0, 0, 1, 0], [1, 0, 1, 1, 1, 0, 1, 1, 0], [0, 0, 1, 1, 0, 0, 1, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 0, 1, 1, 1, 1]]", 3]
["[[1, 0, 1, 0, 0, 1, 0, 1, 0], [1, 1, 1, 0, 1, 1, 0, 0, 1], [1, 1, 1, 1, 1, 0, 0, 1, 0], [1, 0, 1, 1, 1, 0, 1, 1, 0], [0, 0, 1, 1, 0, 0, 1, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 0, 1, 1, 1, 1]]", [0, 6], [8, 2], 3]
["[[1, 0, 1, 0, 0, 1, 0, 1, 0], [1, 1, 1, 0, 1, 1, 0, 0, 1], [1, 1, 1, 1, 1, 0, 0, 1, 0], [1, 0, 1, 1, 1, 0, 1, 1, 0], [0, 0, 1, 1, 0, 0, 1, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 0, 1, 1, 1, 1]]", "(0, 6)", "(8, 2)", "3"]
6
Given 5 labeled water jugs with capacities 76, 135, 134, 19, 10, 30 liters, we aim to fill 3 unlabeled buckets, numbered 1 to 3 and arranged in a line in ascending order, with 390, 420, 439 liters of water respectively. The amount of water in each unlabeled bucket can not at any point in time exceed the amount of water in the bucket placed after it. Jugs can only be filled to the top and emptied completely, and the unlabeled buckets cannot be overfilled. An action, represented as a tuple ('+', X, Y) or ('-', X, Y), involves adding to or removing water from the unlabeled bucket numbered Y, using the jug with capacity X. Determine the shortest sequence of actions needed to fill the buckets as specified, and present the solution as a list of action tuples in Python syntax.
water_jug
subset_sum
4
[["+", 76, 3], ["+", 76, 3], ["+", 134, 3], ["+", 19, 3], ["+", 134, 3], ["+", 134, 2], ["+", 134, 2], ["+", 76, 2], ["+", 76, 2], ["+", 76, 1], ["+", 76, 1], ["+", 134, 1], ["-", 30, 1], ["+", 134, 1]]
14
0.05055570602416992
14
36
3
[[76, 135, 134, 19, 10, 30], [390, 420, 439]]
[[76, 135, 134, 19, 10, 30], [390, 420, 439]]
["[76, 135, 134, 19, 10, 30]", "[390, 420, 439]"]
7
In the 8-puzzle game, you are given a grid with numbered square tiles arranged randomly and one tile missing. The goal is to arrange the tiles in descending order by sliding them into the empty space. The tiles can move in 4 directions: left, right, up, and down. Given the initial state of the puzzle below, where the empty spot is represented as “_”, provide the shortest list of tiles that need to be swapped with the empty spot to achieve the goal state. The goal state is when all tiles are in descending order, with the largest number in the top left corner, and the empty spot is in the bottom right corner. The solution should be a list of numbers in Python format, where each number represents the number on the tile that the empty spot is swapped with at each turn. Initial state of the puzzle: [[36, 16, 48], [86, '_', 13], [89, 31, 37]]
8_puzzle
puzzle
3
[31, 37, 13, 31, 16, 36, 86, 89, 37, 16, 36, 86, 89, 37, 16, 13]
16
0.03079676628112793
16
4
9
[[[36, 16, 48], [86, "_", 13], [89, 31, 37]]]
[[[36, 16, 48], [86, "_", 13], [89, 31, 37]]]
["[[36, 16, 48], [86, '_', 13], [89, 31, 37]]"]
7
In the game 'Sort the Chars', we are given a table of n by m dimensions. This table contains n words, each with m characters, except for the first word which has m - 1 characters. Each character is written on a separate tile. The objective of the game is to rearrange the characters such that row i spells the i-th word in the list, with the blank tile ('_') placed in the top left corner of the board in the end. We can rearrange the tiles by swapping the blank space with any of its 4 diagonal neighboring tiles. Given the list of words and initial state of the board below, where the black space is represented as '_', what is the shortest list of swap actions (reported in python syntax) that can sort the board into the given list of target words? The list must only include the 4 diagonal swap directions: up-right, down-right, up-left, or down-left, representing the direction in ehich the blank space was swpped in. Target words: cry, dune, rump, knee The initial board: [['u', 'c', 'r', 'y'], ['d', 'n', 'n', 'r'], ['_', 'u', 'm', 'p'], ['k', 'e', 'e', 'e']]
8_puzzle_words
puzzle
1
["up-right", "down-right", "up-right", "up-left", "down-left", "down-right", "down-left", "up-left", "up-right", "up-right", "down-right", "down-left", "up-left", "up-left"]
14
0.1806955337524414
14
4
16
[[["u", "c", "r", "y"], ["d", "n", "n", "r"], ["_", "u", "m", "p"], ["k", "e", "e", "e"]]]
[[["u", "c", "r", "y"], ["d", "n", "n", "r"], ["_", "u", "m", "p"], ["k", "e", "e", "e"]], ["cry", "dune", "rump", "knee"]]
["[['u', 'c', 'r', 'y'], ['d', 'n', 'n', 'r'], ['_', 'u', 'm', 'p'], ['k', 'e', 'e', 'e']]", "['cry', 'dune', 'rump', 'knee']"]
7
We have a map of cities, each represented by a letter, and they are connected by one-way roads. The adjacency matrix below shows the connections between the cities. Each row and column represents a city, and a '1' signifies a direct road from the city of the row to the city of the column. The travel time between any two directly connected cities is the same. Currently, we are located in city 'Z'. Our task is to visit city B and city H excatly twice. Determine the quickest route that allows us to visit both these destination cities, ensuring that we stop at the two destinations twice on our path. The sequence in which we visit the destination cities is not important. However, apart from H and B, we can only visit each city once on our path. Provide the solution as a list of the city names on our path, including the start, in Python syntax. R B F Z N H A P M J R 0 1 1 0 1 0 0 0 0 0 B 0 0 1 0 0 0 1 0 0 0 F 0 0 0 1 1 0 0 0 1 0 Z 0 0 0 0 0 0 1 0 0 1 N 0 1 0 0 0 0 0 0 0 0 H 0 0 0 0 1 0 0 0 0 1 A 1 0 0 0 1 0 0 1 0 0 P 0 0 1 1 0 1 0 0 1 0 M 1 0 0 0 0 1 1 0 0 0 J 1 0 0 1 0 0 1 0 0 0
city_directed_graph
pathfinding
10
["Z", "A", "P", "H", "J", "R", "B", "F", "M", "H", "N", "B"]
12
0.02784132957458496
12
10
13
[[[0, 1, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 1], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 1], [1, 0, 0, 0, 1, 0, 0, 1, 0, 0], [0, 0, 1, 1, 0, 1, 0, 0, 1, 0], [1, 0, 0, 0, 0, 1, 1, 0, 0, 0], [1, 0, 0, 1, 0, 0, 1, 0, 0, 0]], ["R", "B", "F", "Z", "N", "H", "A", "P", "M", "J"], "B", "H"]
[[[0, 1, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 1], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 1], [1, 0, 0, 0, 1, 0, 0, 1, 0, 0], [0, 0, 1, 1, 0, 1, 0, 0, 1, 0], [1, 0, 0, 0, 0, 1, 1, 0, 0, 0], [1, 0, 0, 1, 0, 0, 1, 0, 0, 0]], ["R", "B", "F", "Z", "N", "H", "A", "P", "M", "J"], "Z", "B", "H"]
["[[0, 1, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 1], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 1], [1, 0, 0, 0, 1, 0, 0, 1, 0, 0], [0, 0, 1, 1, 0, 1, 0, 0, 1, 0], [1, 0, 0, 0, 0, 1, 1, 0, 0, 0], [1, 0, 0, 1, 0, 0, 1, 0, 0, 0]]", "['R', 'B', 'F', 'Z', 'N', 'H', 'A', 'P', 'M', 'J']", "['Z']", "['B', 'H']"]
7
In the 'taxed coin exchange' problem, you are required to choose a subset of coins from this list [41, 4, 22, 8, 5, 5, 20, 4, 3, 19, 4, 13, 3, 18, 20, 9, 15, 3, 3, 14, 3, 14, 9, 4, 15, 19, 6, 15, 22, 19, 3, 3, 20, 9, 20, 11, 22, 15, 20], such that the sum of the chosen coins adds up to 220. Each coin in the list is unique and can only be used once. Also coins carry a tax value. The tax values for each coin is {20: 11, 15: 13, 9: 7, 8: 3, 3: 1, 6: 3, 5: 2, 13: 11, 4: 4, 22: 11, 19: 19, 11: 1, 14: 2, 41: 16, 18: 18}, where the tax for coins of the same value is the same. Also, if the coin chosen is smaller than the previous one, it must have an even value, otherwise, if the coin is larger than or equal to the previous coin chosen, it must have an odd value. The objective is to determine which subset of coins should be selected to minimize the total tax paid. The solution should be presented as a list of numbers, representing the value of the coins chosen in order, with the first coins chosen being in index 0, formatted in Python syntax.
coin_exchange
subset_sum
21
[14, 8, 6, 11, 41, 22, 20, 14, 4, 5, 5, 9, 9, 9, 13, 15, 15]
115
0.03900480270385742
17
39
39
[[41, 4, 22, 8, 5, 5, 20, 4, 3, 19, 4, 13, 3, 18, 20, 9, 15, 3, 3, 14, 3, 14, 9, 4, 15, 19, 6, 15, 22, 19, 3, 3, 20, 9, 20, 11, 22, 15, 20]]
[[41, 4, 22, 8, 5, 5, 20, 4, 3, 19, 4, 13, 3, 18, 20, 9, 15, 3, 3, 14, 3, 14, 9, 4, 15, 19, 6, 15, 22, 19, 3, 3, 20, 9, 20, 11, 22, 15, 20], {"20": 11, "15": 13, "9": 7, "8": 3, "3": 1, "6": 3, "5": 2, "13": 11, "4": 4, "22": 11, "19": 19, "11": 1, "14": 2, "41": 16, "18": 18}, 220]
["[41, 4, 22, 8, 5, 5, 20, 4, 3, 19, 4, 13, 3, 18, 20, 9, 15, 3, 3, 14, 3, 14, 9, 4, 15, 19, 6, 15, 22, 19, 3, 3, 20, 9, 20, 11, 22, 15, 20]", "{20: 11, 15: 13, 9: 7, 8: 3, 3: 1, 6: 3, 5: 2, 13: 11, 4: 4, 22: 11, 19: 19, 11: 1, 14: 2, 41: 16, 18: 18}", "220"]
7
The game of 'Sort It' begins with 3 tubes, each filled with 4 balls of different colors. The goal is to sort the balls by color, with each tube containing balls of only one color. Only one ball can be moved at a time, taken from the top of one tube and placed on top of another. The capacity of each tube (maximum number of balls we can fit in each tube) is 6 balls. It is not allowed to place a ball in a tube that already has 6 balls. The solution should be a list of tuples, each containing, first, the index of the tube from which a ball is taken and, second, the index of the tube to which it is moved, indexing from 0. Given the initial state of the tubes, represented by the lists below (with the leftmost item being the color of the topmost ball in each tube), what is the shortest list of move tuples that will result in all the balls being correctly sorted? [['Red', 'Red', 'Blue', 'Green'], ['Blue', 'Red', 'Blue', 'Green'], ['Blue', 'Green', 'Red', 'Green']]
color_sorting
sorting
7
[[1, 2], [1, 0], [1, 2], [1, 0], [2, 1], [2, 1], [2, 1], [0, 1], [2, 1], [2, 0], [2, 1], [0, 2], [0, 2], [0, 2], [0, 2], [0, 2], [1, 0], [1, 0], [1, 0], [2, 1]]
20
2.5152063369750977
20
6
12
[[["Red", "Red", "Blue", "Green"], ["Blue", "Red", "Blue", "Green"], ["Blue", "Green", "Red", "Green"]], 6]
[[["Red", "Red", "Blue", "Green"], ["Blue", "Red", "Blue", "Green"], ["Blue", "Green", "Red", "Green"]], 6]
["[['Red', 'Red', 'Blue', 'Green'], ['Blue', 'Red', 'Blue', 'Green'], ['Blue', 'Green', 'Red', 'Green']]", "6"]
7
We have a 3x3 numerical grid, with numbers ranging from 25 to 58 (25 included in the range but 58 is not included). The numbers in each row and column must be strictly increasing or decreasing. This means that either first > second > third or first < second < third in each row and column. If a grid cell is marked with an 'x', the number in that position is hidden. The objective is to replace the 'x's with unique integers from the given range, ensuring that each number only appears once in the grid. The replacements must maintain the consecutive order in each row and column. Additionally, the sum of the numbers in the topmost row plus the numbers in the rightmost column plus the numbers in the diagonal connecting the top-left corner of the grid to its bottom-right corner should be minimized. The solution should be given as a list of tuples in Python syntax. Each tuple should represent the replacement of a number with an 'x' number and contain three elements: the row index of the 'x', the column index of the 'x' (both starting from 0), and the value of the number that replaces the 'x'. The initial state of the grid is as follows: Grid: [['x' 'x' '42'] ['x' '46' 'x'] ['x' 'x' '55']]
consecutive_grid
underdetermined_system
9
[[0, 0, 25], [0, 1, 26], [1, 0, 47], [1, 2, 43], [2, 0, 48], [2, 1, 49]]
359
0.5126192569732666
6
33
9
["[['', '', '42'], ['', '46', ''], ['', '', '55']]", 25, 58]
["[['', '', '42'], ['', '46', ''], ['', '', '55']]", 25, 58]
["[['', '', '42'], ['', '46', ''], ['', '', '55']]", "25", "58"]
7
In the magic square problem, a 3x3 grid is filled with unique integers ranging from 19 to 43. Some numbers are already given, while others are unknown and represented as 'x'. Sum of column 1 (counting from 0) must be 111, and sum of row 1 must be 94. Also, the sum of the numbers in the diagonal from the top right to the bottom left corner of the grid should equal 82. The goal is to find unique integers (ie each number can be in the final grid only once) in the given range to replace with ‘x’s in the grid below such that the sum of the specified rows, columns, and diagonal equals the given amounts and the sum of all of the numbers in the grid is as low as possible. The solution should be provided as a list of tuples in Python syntax. Each tuple should contain three numbers for each 'x' position: the row index, the column index (both starting from 0), and the value of the unique integer replaced with 'x' at that position. Grid: [['37' 'x' '26'] ['x' 'x' 'x'] ['x' 'x' 'x']]
magic_square
underdetermined_system
7
[[0, 1, 33], [1, 0, 23], [1, 1, 36], [1, 2, 35], [2, 0, 20], [2, 1, 42], [2, 2, 19]]
271
1.7812092304229736
7
19
9
["[['37', '', '26'], ['', '', ''], ['', '', '']]", 3, 19, 43]
["[['37', '', '26'], ['', '', ''], ['', '', '']]", 19, 43, [1, 2], [1, 2], [111], [94], 82]
["[['37', '', '26'], ['', '', ''], ['', '', '']]", "19", "43", "[None, 111, None]", "[None, 94, None]", "82"]
7
In 'Restricted Sorting', there are 6 stacks each with a capacity of 4 blocks, with 3 stacks filled with blocks of varying shades and the remaining are empty. The challenge is to sort the blocks by shade such that if a stack is not empty, it is stacked with 3 blocks of a single shade. The player can only transfer one block at a time from the top of a stack to an empty stack or to a stack that has only blocks of that shade, without exceeding the stacks’ capacity. Transferring blocks to certain stacks is more expensive than others. The cost of moving one block to the top of each stack is: {0: 5, 1: 3, 2: 5, 3: 6, 4: 5, 5: 7}, where the keys are the index of each stack, indexing from 0. The cost of moving a block is always at least 1. The solution should be a list of tuples, each containing, first, the index of the stack from which a block is picked up from and, second, the index of the stack to which it is transferred, indexing from 0. Given the initial state of the stacks, represented by the lists below (with the leftmost item being the shade of the topmost block in each stack)(and the first stack being the stack at index 0), what is the list of transfer pairs (reported in python syntax) with the least possible cost, that will result in all the blocks being correctly sorted? [[], [], [], ['Yellow', 'Green', 'Blue', 'Blue'], ['Red', 'Blue', 'Green', 'Green'], ['Red', 'Red', 'Yellow', 'Yellow']]
restricted_sorting
sorting
1
[[4, 1], [5, 1], [5, 1], [3, 0], [3, 2], [4, 3], [0, 5], [2, 4]]
37
0.026184558868408203
8
30
12
[[[], [], [], ["Yellow", "Green", "Blue", "Blue"], ["Red", "Blue", "Green", "Green"], ["Red", "Red", "Yellow", "Yellow"]], 4, {"0": 5, "1": 3, "2": 5, "3": 6, "4": 5, "5": 7}]
[[[], [], [], ["Yellow", "Green", "Blue", "Blue"], ["Red", "Blue", "Green", "Green"], ["Red", "Red", "Yellow", "Yellow"]], 4, {"0": 5, "1": 3, "2": 5, "3": 6, "4": 5, "5": 7}, 3]
["[[], [], [], ['Yellow', 'Green', 'Blue', 'Blue'], ['Red', 'Blue', 'Green', 'Green'], ['Red', 'Red', 'Yellow', 'Yellow']]", "{0: 5, 1: 3, 2: 5, 3: 6, 4: 5, 5: 7}", "4", "3"]
7
Using the provided matrix map of a city, where numbers represent travel time in minutes (all numbers are positive integers) and 'x' marks closed workshops, find the quickest route for Ben to travel from his current workshop at index (6, 1) to his destination workshop at index (2, 6), indexing from 0. Ben's car can move north, south, east, or west from a given crossroad, provided there's no x in that direction. Also, there are 3 districts in the city with district 1 covering rows 0 to 2, district 2 covering rows 3 to 5, and district 3 covering rows 6 to 8. Ben has to visit at least 1 workshop in each district on his path to the destination. The roads are bidirectional. The answer should be a list of tuples (in Python syntax) indicating the index of workshops on Ben's path. The start and end workshops must be included in the path. [x 7 7 12 7 x 13 1 18] [9 18 x 4 19 8 x 15 10] [9 x 10 5 5 9 8 x x] [14 4 19 9 14 x x 7 9] [9 4 11 15 7 x 17 3 x] [11 x 2 18 15 15 x x x] [x 5 10 x x 13 x 9 x] [3 x 2 x 12 x x x 4] [x x 7 x 19 x x x 9]
traffic
pathfinding
1
[[6, 1], [6, 2], [5, 2], [4, 2], [4, 3], [3, 3], [2, 3], [2, 4], [2, 5], [2, 6]]
74
0.021251201629638672
10
4
4
[[["x", "7", "7", "12", "7", "x", "13", "1", "18"], ["9", "18", "x", "4", "19", "8", "x", "15", "10"], ["9", "x", "10", "5", "5", "9", "8", "x", "x"], ["14", "4", "19", "9", "14", "x", "x", "7", "9"], ["9", "4", "11", "15", "7", "x", "17", "3", "x"], ["11", "x", "2", "18", "15", "15", "x", "x", "x"], ["x", "5", "10", "x", "x", "13", "x", "9", "x"], ["3", "x", "2", "x", "12", "x", "x", "x", "4"], ["x", "x", "7", "x", "19", "x", "x", "x", "9"]]]
[[["x", "7", "7", "12", "7", "x", "13", "1", "18"], ["9", "18", "x", "4", "19", "8", "x", "15", "10"], ["9", "x", "10", "5", "5", "9", "8", "x", "x"], ["14", "4", "19", "9", "14", "x", "x", "7", "9"], ["9", "4", "11", "15", "7", "x", "17", "3", "x"], ["11", "x", "2", "18", "15", "15", "x", "x", "x"], ["x", "5", "10", "x", "x", "13", "x", "9", "x"], ["3", "x", "2", "x", "12", "x", "x", "x", "4"], ["x", "x", "7", "x", "19", "x", "x", "x", "9"]], [6, 1], [2, 6], 2, 5]
["[['x', '7', '7', '12', '7', 'x', '13', '1', '18'], ['9', '18', 'x', '4', '19', '8', 'x', '15', '10'], ['9', 'x', '10', '5', '5', '9', '8', 'x', 'x'], ['14', '4', '19', '9', '14', 'x', 'x', '7', '9'], ['9', '4', '11', '15', '7', 'x', '17', '3', 'x'], ['11', 'x', '2', '18', '15', '15', 'x', 'x', 'x'], ['x', '5', '10', 'x', 'x', '13', 'x', '9', 'x'], ['3', 'x', '2', 'x', '12', 'x', 'x', 'x', '4'], ['x', 'x', '7', 'x', '19', 'x', 'x', 'x', '9']]", "(6, 1)", "(2, 6)", "2", "5"]
7
Alex is at a trampoline park with a grid of mini trampolines, arranged in a square of 9x9. Some trampolines are broken and unusable. A map of the park is provided below, with 1 indicating a broken trampoline and 0 indicating a functional one. Alex can jump to any of the eight adjacent trampolines, as long as they are not broken. However, Alex must make excatly 3 diagonal jumps, no more, no less, on his path to his destination. He is currently on the trampoline at position (8, 2) (positions are counted from 0, left to right, top to bottom) and wants to reach the trampoline at position (0, 7). What is the shortest sequence of trampolines he should jump on to reach his destination (including the first and final trampolines)? The answer should be a list of tuples, in Python syntax, indicating the row and column of each trampoline Alex jumps on. 1 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0
trampoline_matrix
pathfinding
9
[[8, 2], [8, 3], [8, 4], [8, 5], [8, 6], [7, 7], [6, 8], [5, 8], [4, 8], [3, 8], [2, 8], [1, 7], [0, 7]]
13
0.030285120010375977
13
8
2
["[[1, 0, 1, 1, 1, 1, 1, 0, 0], [0, 1, 1, 1, 1, 1, 1, 0, 0], [1, 1, 1, 1, 0, 0, 0, 1, 0], [1, 1, 1, 0, 0, 1, 1, 0, 0], [0, 0, 1, 0, 0, 1, 1, 0, 0], [1, 1, 0, 0, 1, 0, 1, 0, 0], [0, 1, 1, 1, 1, 1, 1, 1, 0], [1, 0, 0, 0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0]]", 3]
["[[1, 0, 1, 1, 1, 1, 1, 0, 0], [0, 1, 1, 1, 1, 1, 1, 0, 0], [1, 1, 1, 1, 0, 0, 0, 1, 0], [1, 1, 1, 0, 0, 1, 1, 0, 0], [0, 0, 1, 0, 0, 1, 1, 0, 0], [1, 1, 0, 0, 1, 0, 1, 0, 0], [0, 1, 1, 1, 1, 1, 1, 1, 0], [1, 0, 0, 0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0]]", [8, 2], [0, 7], 3]
["[[1, 0, 1, 1, 1, 1, 1, 0, 0], [0, 1, 1, 1, 1, 1, 1, 0, 0], [1, 1, 1, 1, 0, 0, 0, 1, 0], [1, 1, 1, 0, 0, 1, 1, 0, 0], [0, 0, 1, 0, 0, 1, 1, 0, 0], [1, 1, 0, 0, 1, 0, 1, 0, 0], [0, 1, 1, 1, 1, 1, 1, 1, 0], [1, 0, 0, 0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0]]", "(8, 2)", "(0, 7)", "3"]
7
Given 5 labeled water jugs with capacities 22, 48, 29, 111, 92, 28 liters, we aim to fill 3 unlabeled buckets, numbered 1 to 3 and arranged in a line in ascending order, with 164, 233, 289 liters of water respectively. The amount of water in each unlabeled bucket can not at any point in time exceed the amount of water in the bucket placed after it. Jugs can only be filled to the top and emptied completely, and the unlabeled buckets cannot be overfilled. An action, represented as a tuple ('+', X, Y) or ('-', X, Y), involves adding to or removing water from the unlabeled bucket numbered Y, using the jug with capacity X. Determine the shortest sequence of actions needed to fill the buckets as specified, and present the solution as a list of action tuples in Python syntax.
water_jug
subset_sum
4
[["+", 48, 3], ["+", 111, 3], ["+", 111, 3], ["-", 29, 3], ["+", 48, 3], ["+", 29, 2], ["+", 92, 2], ["+", 92, 2], ["-", 28, 2], ["+", 48, 2], ["+", 22, 1], ["+", 22, 1], ["+", 28, 1], ["+", 92, 1]]
14
0.03762245178222656
14
36
3
[[22, 48, 29, 111, 92, 28], [164, 233, 289]]
[[22, 48, 29, 111, 92, 28], [164, 233, 289]]
["[22, 48, 29, 111, 92, 28]", "[164, 233, 289]"]
8
In the 8-puzzle game, you are given a grid with numbered square tiles arranged randomly and one tile missing. The goal is to arrange the tiles in descending order by sliding them into the empty space. The tiles can move in 4 directions: left, right, up, and down. Given the initial state of the puzzle below, where the empty spot is represented as “_”, provide the shortest list of tiles that need to be swapped with the empty spot to achieve the goal state. The goal state is when all tiles are in descending order, with the largest number in the top left corner, and the empty spot is in the bottom right corner. The solution should be a list of numbers in Python format, where each number represents the number on the tile that the empty spot is swapped with at each turn. Initial state of the puzzle: [[77, 49, '_'], [56, 72, 74], [96, 33, 59]]
8_puzzle
puzzle
3
[74, 72, 56, 96, 33, 56, 49, 77, 96, 49, 72, 59, 56, 33, 49, 72, 59, 56]
18
0.03490924835205078
18
4
9
[[[77, 49, "_"], [56, 72, 74], [96, 33, 59]]]
[[[77, 49, "_"], [56, 72, 74], [96, 33, 59]]]
["[[77, 49, '_'], [56, 72, 74], [96, 33, 59]]"]
8
In the game 'Sort the Chars', we are given a table of n by m dimensions. This table contains n words, each with m characters, except for the first word which has m - 1 characters. Each character is written on a separate tile. The objective of the game is to rearrange the characters such that row i spells the i-th word in the list, with the blank tile ('_') placed in the top left corner of the board in the end. We can rearrange the tiles by swapping the blank space with any of its 4 diagonal neighboring tiles. Given the list of words and initial state of the board below, where the black space is represented as '_', what is the shortest list of swap actions (reported in python syntax) that can sort the board into the given list of target words? The list must only include the 4 diagonal swap directions: up-right, down-right, up-left, or down-left, representing the direction in ehich the blank space was swpped in. Target words: ken, khat, kill, went The initial board: [['h', 'k', 't', 'n'], ['k', 'e', 'a', 'k'], ['e', 'i', '_', 'l'], ['w', 'l', 'n', 't']]
8_puzzle_words
puzzle
1
["up-right", "up-left", "down-left", "down-right", "down-left", "up-left", "up-right", "up-left"]
8
0.16104960441589355
8
4
16
[[["h", "k", "t", "n"], ["k", "e", "a", "k"], ["e", "i", "_", "l"], ["w", "l", "n", "t"]]]
[[["h", "k", "t", "n"], ["k", "e", "a", "k"], ["e", "i", "_", "l"], ["w", "l", "n", "t"]], ["ken", "khat", "kill", "went"]]
["[['h', 'k', 't', 'n'], ['k', 'e', 'a', 'k'], ['e', 'i', '_', 'l'], ['w', 'l', 'n', 't']]", "['ken', 'khat', 'kill', 'went']"]
8
We have a map of cities, each represented by a letter, and they are connected by one-way roads. The adjacency matrix below shows the connections between the cities. Each row and column represents a city, and a '1' signifies a direct road from the city of the row to the city of the column. The travel time between any two directly connected cities is the same. Currently, we are located in city 'U'. Our task is to visit city A and city X excatly twice. Determine the quickest route that allows us to visit both these destination cities, ensuring that we stop at the two destinations twice on our path. The sequence in which we visit the destination cities is not important. However, apart from X and A, we can only visit each city once on our path. Provide the solution as a list of the city names on our path, including the start, in Python syntax. Y O Q X K R A U M L Y 0 1 0 0 0 1 1 1 1 1 O 0 0 0 0 0 0 1 0 0 0 Q 1 0 0 1 0 1 0 0 1 0 X 1 0 1 0 0 1 0 0 1 1 K 0 1 0 1 0 0 1 0 0 0 R 0 0 0 0 1 0 0 0 0 0 A 0 0 0 1 0 1 0 0 1 1 U 0 0 0 0 0 0 0 0 0 1 M 0 0 1 0 0 0 0 0 0 0 L 0 1 0 0 0 0 0 0 0 0
city_directed_graph
pathfinding
10
["U", "L", "O", "A", "X", "Y", "A", "X"]
8
0.029206514358520508
8
10
13
[[[0, 1, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [1, 0, 0, 1, 0, 1, 0, 0, 1, 0], [1, 0, 1, 0, 0, 1, 0, 0, 1, 1], [0, 1, 0, 1, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 1, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0]], ["Y", "O", "Q", "X", "K", "R", "A", "U", "M", "L"], "A", "X"]
[[[0, 1, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [1, 0, 0, 1, 0, 1, 0, 0, 1, 0], [1, 0, 1, 0, 0, 1, 0, 0, 1, 1], [0, 1, 0, 1, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 1, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0]], ["Y", "O", "Q", "X", "K", "R", "A", "U", "M", "L"], "U", "A", "X"]
["[[0, 1, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [1, 0, 0, 1, 0, 1, 0, 0, 1, 0], [1, 0, 1, 0, 0, 1, 0, 0, 1, 1], [0, 1, 0, 1, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 1, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0]]", "['Y', 'O', 'Q', 'X', 'K', 'R', 'A', 'U', 'M', 'L']", "['U']", "['A', 'X']"]
8
In the 'taxed coin exchange' problem, you are required to choose a subset of coins from this list [4, 17, 8, 15, 6, 14, 2, 4, 16, 10, 6, 5, 14, 8, 18, 2, 17, 9, 17, 59, 3, 10, 21, 4, 9, 14, 14, 20, 14, 8, 14, 18, 5, 16, 14, 17, 5, 3], such that the sum of the chosen coins adds up to 214. Each coin in the list is unique and can only be used once. Also coins carry a tax value. The tax values for each coin is {15: 6, 5: 5, 17: 1, 14: 4, 2: 2, 8: 2, 10: 9, 59: 9, 16: 8, 4: 3, 18: 10, 20: 11, 21: 8, 6: 6, 9: 9, 3: 3}, where the tax for coins of the same value is the same. Also, if the coin chosen is smaller than the previous one, it must have an even value, otherwise, if the coin is larger than or equal to the previous coin chosen, it must have an odd value. The objective is to determine which subset of coins should be selected to minimize the total tax paid. The solution should be presented as a list of numbers, representing the value of the coins chosen in order, with the first coins chosen being in index 0, formatted in Python syntax.
coin_exchange
subset_sum
22
[17, 17, 17, 14, 8, 59, 14, 8, 17, 14, 8, 21]
39
0.037581682205200195
12
38
38
[[4, 17, 8, 15, 6, 14, 2, 4, 16, 10, 6, 5, 14, 8, 18, 2, 17, 9, 17, 59, 3, 10, 21, 4, 9, 14, 14, 20, 14, 8, 14, 18, 5, 16, 14, 17, 5, 3]]
[[4, 17, 8, 15, 6, 14, 2, 4, 16, 10, 6, 5, 14, 8, 18, 2, 17, 9, 17, 59, 3, 10, 21, 4, 9, 14, 14, 20, 14, 8, 14, 18, 5, 16, 14, 17, 5, 3], {"15": 6, "5": 5, "17": 1, "14": 4, "2": 2, "8": 2, "10": 9, "59": 9, "16": 8, "4": 3, "18": 10, "20": 11, "21": 8, "6": 6, "9": 9, "3": 3}, 214]
["[4, 17, 8, 15, 6, 14, 2, 4, 16, 10, 6, 5, 14, 8, 18, 2, 17, 9, 17, 59, 3, 10, 21, 4, 9, 14, 14, 20, 14, 8, 14, 18, 5, 16, 14, 17, 5, 3]", "{15: 6, 5: 5, 17: 1, 14: 4, 2: 2, 8: 2, 10: 9, 59: 9, 16: 8, 4: 3, 18: 10, 20: 11, 21: 8, 6: 6, 9: 9, 3: 3}", "214"]
8
The game of 'Sort It' begins with 3 tubes, each filled with 4 balls of different colors. The goal is to sort the balls by color, with each tube containing balls of only one color. Only one ball can be moved at a time, taken from the top of one tube and placed on top of another. The capacity of each tube (maximum number of balls we can fit in each tube) is 6 balls. It is not allowed to place a ball in a tube that already has 6 balls. The solution should be a list of tuples, each containing, first, the index of the tube from which a ball is taken and, second, the index of the tube to which it is moved, indexing from 0. Given the initial state of the tubes, represented by the lists below (with the leftmost item being the color of the topmost ball in each tube), what is the shortest list of move tuples that will result in all the balls being correctly sorted? [['Green', 'Blue', 'Red', 'Red'], ['Red', 'Green', 'Green', 'Blue'], ['Red', 'Green', 'Blue', 'Blue']]
color_sorting
sorting
7
[[1, 2], [1, 0], [1, 0], [1, 2], [0, 1], [0, 1], [0, 1], [0, 1], [2, 1], [2, 0], [2, 0], [2, 0], [1, 2], [1, 2], [0, 1]]
15
0.3737452030181885
15
6
12
[[["Green", "Blue", "Red", "Red"], ["Red", "Green", "Green", "Blue"], ["Red", "Green", "Blue", "Blue"]], 6]
[[["Green", "Blue", "Red", "Red"], ["Red", "Green", "Green", "Blue"], ["Red", "Green", "Blue", "Blue"]], 6]
["[['Green', 'Blue', 'Red', 'Red'], ['Red', 'Green', 'Green', 'Blue'], ['Red', 'Green', 'Blue', 'Blue']]", "6"]
8
We have a 3x3 numerical grid, with numbers ranging from 19 to 52 (19 included in the range but 52 is not included). The numbers in each row and column must be strictly increasing or decreasing. This means that either first > second > third or first < second < third in each row and column. If a grid cell is marked with an 'x', the number in that position is hidden. The objective is to replace the 'x's with unique integers from the given range, ensuring that each number only appears once in the grid. The replacements must maintain the consecutive order in each row and column. Additionally, the sum of the numbers in the topmost row plus the numbers in the rightmost column plus the numbers in the diagonal connecting the top-left corner of the grid to its bottom-right corner should be minimized. The solution should be given as a list of tuples in Python syntax. Each tuple should represent the replacement of a number with an 'x' number and contain three elements: the row index of the 'x', the column index of the 'x' (both starting from 0), and the value of the number that replaces the 'x'. The initial state of the grid is as follows: Grid: [['x' 'x' 'x'] ['x' '36' '44'] ['x' '34' 'x']]
consecutive_grid
underdetermined_system
9
[[0, 0, 19], [0, 1, 37], [0, 2, 45], [1, 0, 21], [2, 0, 35], [2, 2, 20]]
285
6.972386598587036
6
33
9
["[['', '', ''], ['', '36', '44'], ['', '34', '']]", 19, 52]
["[['', '', ''], ['', '36', '44'], ['', '34', '']]", 19, 52]
["[['', '', ''], ['', '36', '44'], ['', '34', '']]", "19", "52"]
8
In the magic square problem, a 3x3 grid is filled with unique integers ranging from 35 to 59. Some numbers are already given, while others are unknown and represented as 'x'. Sum of column 1 (counting from 0) must be 130, and sum of row 1 must be 144. Also, the sum of the numbers in the diagonal from the top right to the bottom left corner of the grid should equal 123. The goal is to find unique integers (ie each number can be in the final grid only once) in the given range to replace with ‘x’s in the grid below such that the sum of the specified rows, columns, and diagonal equals the given amounts and the sum of all of the numbers in the grid is as low as possible. The solution should be provided as a list of tuples in Python syntax. Each tuple should contain three numbers for each 'x' position: the row index, the column index (both starting from 0), and the value of the unique integer replaced with 'x' at that position. Grid: [['52' 'x' 'x'] ['x' 'x' 'x'] ['x' 'x' 'x']]
magic_square
underdetermined_system
7
[[0, 1, 38], [0, 2, 35], [1, 0, 39], [1, 1, 51], [1, 2, 54], [2, 0, 37], [2, 1, 41], [2, 2, 36]]
383
2.281245708465576
8
19
9
["[['52', '', ''], ['', '', ''], ['', '', '']]", 3, 35, 59]
["[['52', '', ''], ['', '', ''], ['', '', '']]", 35, 59, [1, 2], [1, 2], [130], [144], 123]
["[['52', '', ''], ['', '', ''], ['', '', '']]", "35", "59", "[None, 130, None]", "[None, 144, None]", "123"]
8
In 'Restricted Sorting', there are 6 stacks each with a capacity of 4 blocks, with 3 stacks filled with blocks of varying shades and the remaining are empty. The challenge is to sort the blocks by shade such that if a stack is not empty, it is stacked with 3 blocks of a single shade. The player can only transfer one block at a time from the top of a stack to an empty stack or to a stack that has only blocks of that shade, without exceeding the stacks’ capacity. Transferring blocks to certain stacks is more expensive than others. The cost of moving one block to the top of each stack is: {0: 3, 1: 6, 2: 1, 3: 7, 4: 2, 5: 3}, where the keys are the index of each stack, indexing from 0. The cost of moving a block is always at least 1. The solution should be a list of tuples, each containing, first, the index of the stack from which a block is picked up from and, second, the index of the stack to which it is transferred, indexing from 0. Given the initial state of the stacks, represented by the lists below (with the leftmost item being the shade of the topmost block in each stack)(and the first stack being the stack at index 0), what is the list of transfer pairs (reported in python syntax) with the least possible cost, that will result in all the blocks being correctly sorted? [['Blue', 'Green', 'Green', 'Red'], ['Yellow', 'Yellow', 'Green', 'Yellow'], [], [], [], ['Red', 'Blue', 'Red', 'Blue']]
restricted_sorting
sorting
1
[[0, 2], [0, 4], [0, 4], [5, 0], [5, 2], [5, 0], [2, 5], [2, 5], [1, 2], [1, 2], [1, 4], [1, 2]]
23
0.13733744621276855
12
30
12
[[["Blue", "Green", "Green", "Red"], ["Yellow", "Yellow", "Green", "Yellow"], [], [], [], ["Red", "Blue", "Red", "Blue"]], 4, {"0": 3, "1": 6, "2": 1, "3": 7, "4": 2, "5": 3}]
[[["Blue", "Green", "Green", "Red"], ["Yellow", "Yellow", "Green", "Yellow"], [], [], [], ["Red", "Blue", "Red", "Blue"]], 4, {"0": 3, "1": 6, "2": 1, "3": 7, "4": 2, "5": 3}, 3]
["[['Blue', 'Green', 'Green', 'Red'], ['Yellow', 'Yellow', 'Green', 'Yellow'], [], [], [], ['Red', 'Blue', 'Red', 'Blue']]", "{0: 3, 1: 6, 2: 1, 3: 7, 4: 2, 5: 3}", "4", "3"]
8
Using the provided matrix map of a city, where numbers represent travel time in minutes (all numbers are positive integers) and 'x' marks closed workshops, find the quickest route for Ben to travel from his current workshop at index (2, 8) to his destination workshop at index (6, 3), indexing from 0. Ben's car can move north, south, east, or west from a given crossroad, provided there's no x in that direction. Also, there are 3 districts in the city with district 1 covering rows 0 to 2, district 2 covering rows 3 to 5, and district 3 covering rows 6 to 8. Ben has to visit at least 1 workshop in each district on his path to the destination. The roads are bidirectional. The answer should be a list of tuples (in Python syntax) indicating the index of workshops on Ben's path. The start and end workshops must be included in the path. [x x x x 13 x x 3 8] [x 18 19 19 2 9 x x x] [15 x 6 x x x 13 13 4] [7 x 4 10 x 11 10 17 x] [2 7 x x 8 16 4 1 4] [x x x 5 16 15 x 17 10] [x x x 5 14 9 3 15 x] [x 6 4 x x x 18 x x] [17 x 19 x 1 4 8 x 8]
traffic
pathfinding
1
[[2, 8], [2, 7], [3, 7], [4, 7], [4, 6], [4, 5], [4, 4], [5, 4], [5, 3], [6, 3]]
85
0.026849031448364258
10
4
4
[[["x", "x", "x", "x", "13", "x", "x", "3", "8"], ["x", "18", "19", "19", "2", "9", "x", "x", "x"], ["15", "x", "6", "x", "x", "x", "13", "13", "4"], ["7", "x", "4", "10", "x", "11", "10", "17", "x"], ["2", "7", "x", "x", "8", "16", "4", "1", "4"], ["x", "x", "x", "5", "16", "15", "x", "17", "10"], ["x", "x", "x", "5", "14", "9", "3", "15", "x"], ["x", "6", "4", "x", "x", "x", "18", "x", "x"], ["17", "x", "19", "x", "1", "4", "8", "x", "8"]]]
[[["x", "x", "x", "x", "13", "x", "x", "3", "8"], ["x", "18", "19", "19", "2", "9", "x", "x", "x"], ["15", "x", "6", "x", "x", "x", "13", "13", "4"], ["7", "x", "4", "10", "x", "11", "10", "17", "x"], ["2", "7", "x", "x", "8", "16", "4", "1", "4"], ["x", "x", "x", "5", "16", "15", "x", "17", "10"], ["x", "x", "x", "5", "14", "9", "3", "15", "x"], ["x", "6", "4", "x", "x", "x", "18", "x", "x"], ["17", "x", "19", "x", "1", "4", "8", "x", "8"]], [2, 8], [6, 3], 2, 5]
["[['x', 'x', 'x', 'x', '13', 'x', 'x', '3', '8'], ['x', '18', '19', '19', '2', '9', 'x', 'x', 'x'], ['15', 'x', '6', 'x', 'x', 'x', '13', '13', '4'], ['7', 'x', '4', '10', 'x', '11', '10', '17', 'x'], ['2', '7', 'x', 'x', '8', '16', '4', '1', '4'], ['x', 'x', 'x', '5', '16', '15', 'x', '17', '10'], ['x', 'x', 'x', '5', '14', '9', '3', '15', 'x'], ['x', '6', '4', 'x', 'x', 'x', '18', 'x', 'x'], ['17', 'x', '19', 'x', '1', '4', '8', 'x', '8']]", "(2, 8)", "(6, 3)", "2", "5"]
8
Alex is at a trampoline park with a grid of mini trampolines, arranged in a square of 9x9. Some trampolines are broken and unusable. A map of the park is provided below, with 1 indicating a broken trampoline and 0 indicating a functional one. Alex can jump to any of the eight adjacent trampolines, as long as they are not broken. However, Alex must make excatly 3 diagonal jumps, no more, no less, on his path to his destination. He is currently on the trampoline at position (0, 8) (positions are counted from 0, left to right, top to bottom) and wants to reach the trampoline at position (8, 3). What is the shortest sequence of trampolines he should jump on to reach his destination (including the first and final trampolines)? The answer should be a list of tuples, in Python syntax, indicating the row and column of each trampoline Alex jumps on. 0 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 1 0 1 1 1 1 0 0 1 0 0 1 0 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1
trampoline_matrix
pathfinding
9
[[0, 8], [1, 7], [2, 7], [3, 7], [4, 7], [5, 6], [5, 5], [6, 5], [7, 4], [7, 3], [8, 3]]
11
0.03156876564025879
11
8
2
["[[0, 1, 1, 1, 0, 0, 0, 1, 0], [0, 1, 1, 0, 1, 0, 1, 0, 0], [1, 0, 1, 1, 1, 1, 0, 0, 1], [0, 0, 1, 0, 1, 1, 1, 0, 1], [1, 0, 1, 1, 0, 1, 1, 0, 1], [1, 1, 1, 1, 1, 0, 0, 0, 0], [0, 0, 1, 1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0, 0, 1, 1]]", 3]
["[[0, 1, 1, 1, 0, 0, 0, 1, 0], [0, 1, 1, 0, 1, 0, 1, 0, 0], [1, 0, 1, 1, 1, 1, 0, 0, 1], [0, 0, 1, 0, 1, 1, 1, 0, 1], [1, 0, 1, 1, 0, 1, 1, 0, 1], [1, 1, 1, 1, 1, 0, 0, 0, 0], [0, 0, 1, 1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0, 0, 1, 1]]", [0, 8], [8, 3], 3]
["[[0, 1, 1, 1, 0, 0, 0, 1, 0], [0, 1, 1, 0, 1, 0, 1, 0, 0], [1, 0, 1, 1, 1, 1, 0, 0, 1], [0, 0, 1, 0, 1, 1, 1, 0, 1], [1, 0, 1, 1, 0, 1, 1, 0, 1], [1, 1, 1, 1, 1, 0, 0, 0, 0], [0, 0, 1, 1, 1, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0, 0, 1, 1]]", "(0, 8)", "(8, 3)", "3"]
8
Given 5 labeled water jugs with capacities 59, 68, 104, 106, 69, 42 liters, we aim to fill 3 unlabeled buckets, numbered 1 to 3 and arranged in a line in ascending order, with 196, 210, 238 liters of water respectively. The amount of water in each unlabeled bucket can not at any point in time exceed the amount of water in the bucket placed after it. Jugs can only be filled to the top and emptied completely, and the unlabeled buckets cannot be overfilled. An action, represented as a tuple ('+', X, Y) or ('-', X, Y), involves adding to or removing water from the unlabeled bucket numbered Y, using the jug with capacity X. Determine the shortest sequence of actions needed to fill the buckets as specified, and present the solution as a list of action tuples in Python syntax.
water_jug
subset_sum
4
[["+", 106, 3], ["+", 106, 3], ["-", 42, 3], ["+", 68, 3], ["+", 104, 2], ["+", 106, 2], ["+", 59, 1], ["+", 68, 1], ["+", 69, 1]]
9
0.032889366149902344
9
36
3
[[59, 68, 104, 106, 69, 42], [196, 210, 238]]
[[59, 68, 104, 106, 69, 42], [196, 210, 238]]
["[59, 68, 104, 106, 69, 42]", "[196, 210, 238]"]
9
In the 8-puzzle game, you are given a grid with numbered square tiles arranged randomly and one tile missing. The goal is to arrange the tiles in descending order by sliding them into the empty space. The tiles can move in 4 directions: left, right, up, and down. Given the initial state of the puzzle below, where the empty spot is represented as “_”, provide the shortest list of tiles that need to be swapped with the empty spot to achieve the goal state. The goal state is when all tiles are in descending order, with the largest number in the top left corner, and the empty spot is in the bottom right corner. The solution should be a list of numbers in Python format, where each number represents the number on the tile that the empty spot is swapped with at each turn. Initial state of the puzzle: [[23, 30, 43], [12, 31, 19], [6, 68, '_']]
8_puzzle
puzzle
3
[19, 31, 68, 6, 12, 23, 30, 68, 23, 30, 68, 43, 31, 19]
14
0.02962183952331543
14
4
9
[[[23, 30, 43], [12, 31, 19], [6, 68, "_"]]]
[[[23, 30, 43], [12, 31, 19], [6, 68, "_"]]]
["[[23, 30, 43], [12, 31, 19], [6, 68, '_']]"]
9
In the game 'Sort the Chars', we are given a table of n by m dimensions. This table contains n words, each with m characters, except for the first word which has m - 1 characters. Each character is written on a separate tile. The objective of the game is to rearrange the characters such that row i spells the i-th word in the list, with the blank tile ('_') placed in the top left corner of the board in the end. We can rearrange the tiles by swapping the blank space with any of its 4 diagonal neighboring tiles. Given the list of words and initial state of the board below, where the black space is represented as '_', what is the shortest list of swap actions (reported in python syntax) that can sort the board into the given list of target words? The list must only include the 4 diagonal swap directions: up-right, down-right, up-left, or down-left, representing the direction in ehich the blank space was swpped in. Target words: sun, quot, chat, lure The initial board: [['_', 's', 'a', 'n'], ['q', 'u', 'o', 'u'], ['c', 'h', 'u', 't'], ['l', 't', 'r', 'e']]
8_puzzle_words
puzzle
1
["down-right", "down-right", "down-left", "up-left", "up-right", "down-right", "up-right", "up-left", "down-left", "down-left", "down-right", "up-right", "up-right", "up-left", "down-left", "up-left"]
16
0.14909648895263672
16
4
16
[[["_", "s", "a", "n"], ["q", "u", "o", "u"], ["c", "h", "u", "t"], ["l", "t", "r", "e"]]]
[[["_", "s", "a", "n"], ["q", "u", "o", "u"], ["c", "h", "u", "t"], ["l", "t", "r", "e"]], ["sun", "quot", "chat", "lure"]]
["[['_', 's', 'a', 'n'], ['q', 'u', 'o', 'u'], ['c', 'h', 'u', 't'], ['l', 't', 'r', 'e']]", "['sun', 'quot', 'chat', 'lure']"]
9
We have a map of cities, each represented by a letter, and they are connected by one-way roads. The adjacency matrix below shows the connections between the cities. Each row and column represents a city, and a '1' signifies a direct road from the city of the row to the city of the column. The travel time between any two directly connected cities is the same. Currently, we are located in city 'C'. Our task is to visit city W and city F excatly twice. Determine the quickest route that allows us to visit both these destination cities, ensuring that we stop at the two destinations twice on our path. The sequence in which we visit the destination cities is not important. However, apart from F and W, we can only visit each city once on our path. Provide the solution as a list of the city names on our path, including the start, in Python syntax. R T C F S X Y E A W R 0 1 1 0 0 0 1 1 0 1 T 0 0 1 1 0 0 0 0 0 1 C 0 0 0 0 0 0 1 0 0 0 F 1 1 0 0 0 1 0 1 1 0 S 1 0 0 1 0 0 0 1 0 0 X 1 0 1 1 1 0 0 0 0 0 Y 1 1 0 1 0 0 0 0 0 0 E 0 0 1 0 1 0 0 0 0 0 A 1 0 0 0 1 1 1 1 0 0 W 1 0 0 1 1 1 0 1 1 0
city_directed_graph
pathfinding
10
["C", "Y", "F", "T", "W", "R", "W", "F"]
8
0.022681236267089844
8
10
13
[[[0, 1, 1, 0, 0, 0, 1, 1, 0, 1], [0, 0, 1, 1, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1, 0, 1, 1, 0], [1, 0, 0, 1, 0, 0, 0, 1, 0, 0], [1, 0, 1, 1, 1, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, 1, 1, 1, 0, 0], [1, 0, 0, 1, 1, 1, 0, 1, 1, 0]], ["R", "T", "C", "F", "S", "X", "Y", "E", "A", "W"], "W", "F"]
[[[0, 1, 1, 0, 0, 0, 1, 1, 0, 1], [0, 0, 1, 1, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1, 0, 1, 1, 0], [1, 0, 0, 1, 0, 0, 0, 1, 0, 0], [1, 0, 1, 1, 1, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, 1, 1, 1, 0, 0], [1, 0, 0, 1, 1, 1, 0, 1, 1, 0]], ["R", "T", "C", "F", "S", "X", "Y", "E", "A", "W"], "C", "W", "F"]
["[[0, 1, 1, 0, 0, 0, 1, 1, 0, 1], [0, 0, 1, 1, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [1, 1, 0, 0, 0, 1, 0, 1, 1, 0], [1, 0, 0, 1, 0, 0, 0, 1, 0, 0], [1, 0, 1, 1, 1, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, 1, 1, 1, 0, 0], [1, 0, 0, 1, 1, 1, 0, 1, 1, 0]]", "['R', 'T', 'C', 'F', 'S', 'X', 'Y', 'E', 'A', 'W']", "['C']", "['W', 'F']"]
9
In the 'taxed coin exchange' problem, you are required to choose a subset of coins from this list [3, 6, 9, 10, 13, 15, 18, 5, 21, 19, 12, 15, 5, 9, 4, 16, 8, 4, 7, 7, 7, 2, 16, 14, 18, 3, 89, 21, 12, 10, 7, 14, 4, 11, 6, 20], such that the sum of the chosen coins adds up to 229. Each coin in the list is unique and can only be used once. Also coins carry a tax value. The tax values for each coin is {14: 1, 89: 13, 2: 2, 5: 2, 4: 4, 6: 6, 8: 2, 16: 5, 21: 4, 20: 2, 18: 9, 11: 10, 10: 3, 12: 12, 15: 5, 13: 1, 3: 1, 19: 19, 7: 7, 9: 3}, where the tax for coins of the same value is the same. Also, if the coin chosen is smaller than the previous one, it must have an even value, otherwise, if the coin is larger than or equal to the previous coin chosen, it must have an odd value. The objective is to determine which subset of coins should be selected to minimize the total tax paid. The solution should be presented as a list of numbers, representing the value of the coins chosen in order, with the first coins chosen being in index 0, formatted in Python syntax.
coin_exchange
subset_sum
23
[3, 89, 20, 14, 21, 14, 8, 13, 21, 16, 10]
37
0.03856658935546875
11
36
36
[[3, 6, 9, 10, 13, 15, 18, 5, 21, 19, 12, 15, 5, 9, 4, 16, 8, 4, 7, 7, 7, 2, 16, 14, 18, 3, 89, 21, 12, 10, 7, 14, 4, 11, 6, 20]]
[[3, 6, 9, 10, 13, 15, 18, 5, 21, 19, 12, 15, 5, 9, 4, 16, 8, 4, 7, 7, 7, 2, 16, 14, 18, 3, 89, 21, 12, 10, 7, 14, 4, 11, 6, 20], {"14": 1, "89": 13, "2": 2, "5": 2, "4": 4, "6": 6, "8": 2, "16": 5, "21": 4, "20": 2, "18": 9, "11": 10, "10": 3, "12": 12, "15": 5, "13": 1, "3": 1, "19": 19, "7": 7, "9": 3}, 229]
["[3, 6, 9, 10, 13, 15, 18, 5, 21, 19, 12, 15, 5, 9, 4, 16, 8, 4, 7, 7, 7, 2, 16, 14, 18, 3, 89, 21, 12, 10, 7, 14, 4, 11, 6, 20]", "{14: 1, 89: 13, 2: 2, 5: 2, 4: 4, 6: 6, 8: 2, 16: 5, 21: 4, 20: 2, 18: 9, 11: 10, 10: 3, 12: 12, 15: 5, 13: 1, 3: 1, 19: 19, 7: 7, 9: 3}", "229"]
9
The game of 'Sort It' begins with 3 tubes, each filled with 4 balls of different colors. The goal is to sort the balls by color, with each tube containing balls of only one color. Only one ball can be moved at a time, taken from the top of one tube and placed on top of another. The capacity of each tube (maximum number of balls we can fit in each tube) is 6 balls. It is not allowed to place a ball in a tube that already has 6 balls. The solution should be a list of tuples, each containing, first, the index of the tube from which a ball is taken and, second, the index of the tube to which it is moved, indexing from 0. Given the initial state of the tubes, represented by the lists below (with the leftmost item being the color of the topmost ball in each tube), what is the shortest list of move tuples that will result in all the balls being correctly sorted? [['Green', 'Blue', 'Green', 'Red'], ['Red', 'Green', 'Blue', 'Red'], ['Blue', 'Red', 'Green', 'Blue']]
color_sorting
sorting
7
[[0, 2], [1, 0], [1, 2], [1, 0], [1, 0], [2, 1], [2, 1], [0, 1], [2, 0], [2, 1], [2, 1], [0, 2], [0, 2], [0, 1], [0, 2], [0, 2], [1, 0], [1, 2], [1, 0], [1, 0], [2, 1], [2, 1]]
22
7.09385085105896
22
6
12
[[["Green", "Blue", "Green", "Red"], ["Red", "Green", "Blue", "Red"], ["Blue", "Red", "Green", "Blue"]], 6]
[[["Green", "Blue", "Green", "Red"], ["Red", "Green", "Blue", "Red"], ["Blue", "Red", "Green", "Blue"]], 6]
["[['Green', 'Blue', 'Green', 'Red'], ['Red', 'Green', 'Blue', 'Red'], ['Blue', 'Red', 'Green', 'Blue']]", "6"]
9
We have a 3x3 numerical grid, with numbers ranging from 34 to 67 (34 included in the range but 67 is not included). The numbers in each row and column must be strictly increasing or decreasing. This means that either first > second > third or first < second < third in each row and column. If a grid cell is marked with an 'x', the number in that position is hidden. The objective is to replace the 'x's with unique integers from the given range, ensuring that each number only appears once in the grid. The replacements must maintain the consecutive order in each row and column. Additionally, the sum of the numbers in the topmost row plus the numbers in the rightmost column plus the numbers in the diagonal connecting the top-left corner of the grid to its bottom-right corner should be minimized. The solution should be given as a list of tuples in Python syntax. Each tuple should represent the replacement of a number with an 'x' number and contain three elements: the row index of the 'x', the column index of the 'x' (both starting from 0), and the value of the number that replaces the 'x'. The initial state of the grid is as follows: Grid: [['x' 'x' 'x'] ['x' '51' 'x'] ['x' '61' '64']]
consecutive_grid
underdetermined_system
9
[[0, 0, 34], [0, 1, 35], [0, 2, 36], [1, 0, 52], [1, 2, 37], [2, 0, 53]]
391
0.19122576713562012
6
33
9
["[['', '', ''], ['', '51', ''], ['', '61', '64']]", 34, 67]
["[['', '', ''], ['', '51', ''], ['', '61', '64']]", 34, 67]
["[['', '', ''], ['', '51', ''], ['', '61', '64']]", "34", "67"]
9
In the magic square problem, a 3x3 grid is filled with unique integers ranging from 35 to 59. Some numbers are already given, while others are unknown and represented as 'x'. Sum of column 1 (counting from 0) must be 125, and sum of row 1 must be 157. Also, the sum of the numbers in the diagonal from the top right to the bottom left corner of the grid should equal 138. The goal is to find unique integers (ie each number can be in the final grid only once) in the given range to replace with ‘x’s in the grid below such that the sum of the specified rows, columns, and diagonal equals the given amounts and the sum of all of the numbers in the grid is as low as possible. The solution should be provided as a list of tuples in Python syntax. Each tuple should contain three numbers for each 'x' position: the row index, the column index (both starting from 0), and the value of the unique integer replaced with 'x' at that position. Grid: [['x' 'x' 'x'] ['x' 'x' 'x'] ['37' 'x' 'x']]
magic_square
underdetermined_system
6
[[0, 0, 38], [0, 1, 35], [0, 2, 47], [1, 0, 45], [1, 1, 54], [1, 2, 58], [2, 1, 36], [2, 2, 39]]
389
1.5451362133026123
8
24
9
["[['', '', ''], ['', '', ''], ['37', '', '']]", 3, 35, 59]
["[['', '', ''], ['', '', ''], ['37', '', '']]", 35, 59, [1, 2], [1, 2], [125], [157], 138]
["[['', '', ''], ['', '', ''], ['37', '', '']]", "35", "59", "[None, 125, None]", "[None, 157, None]", "138"]
9
In 'Restricted Sorting', there are 6 stacks each with a capacity of 4 blocks, with 3 stacks filled with blocks of varying shades and the remaining are empty. The challenge is to sort the blocks by shade such that if a stack is not empty, it is stacked with 3 blocks of a single shade. The player can only transfer one block at a time from the top of a stack to an empty stack or to a stack that has only blocks of that shade, without exceeding the stacks’ capacity. Transferring blocks to certain stacks is more expensive than others. The cost of moving one block to the top of each stack is: {0: 7, 1: 1, 2: 4, 3: 7, 4: 4, 5: 4}, where the keys are the index of each stack, indexing from 0. The cost of moving a block is always at least 1. The solution should be a list of tuples, each containing, first, the index of the stack from which a block is picked up from and, second, the index of the stack to which it is transferred, indexing from 0. Given the initial state of the stacks, represented by the lists below (with the leftmost item being the shade of the topmost block in each stack)(and the first stack being the stack at index 0), what is the list of transfer pairs (reported in python syntax) with the least possible cost, that will result in all the blocks being correctly sorted? [['Yellow', 'Yellow', 'Red', 'Red'], [], ['Blue', 'Yellow', 'Red', 'Green'], [], ['Green', 'Blue', 'Green', 'Blue'], []]
restricted_sorting
sorting
1
[[0, 3], [0, 3], [2, 5], [2, 3], [2, 1], [4, 2], [4, 5], [4, 2], [4, 5], [0, 1], [0, 1]]
44
0.111785888671875
11
30
12
[[["Yellow", "Yellow", "Red", "Red"], [], ["Blue", "Yellow", "Red", "Green"], [], ["Green", "Blue", "Green", "Blue"], []], 4, {"0": 7, "1": 1, "2": 4, "3": 7, "4": 4, "5": 4}]
[[["Yellow", "Yellow", "Red", "Red"], [], ["Blue", "Yellow", "Red", "Green"], [], ["Green", "Blue", "Green", "Blue"], []], 4, {"0": 7, "1": 1, "2": 4, "3": 7, "4": 4, "5": 4}, 3]
["[['Yellow', 'Yellow', 'Red', 'Red'], [], ['Blue', 'Yellow', 'Red', 'Green'], [], ['Green', 'Blue', 'Green', 'Blue'], []]", "{0: 7, 1: 1, 2: 4, 3: 7, 4: 4, 5: 4}", "4", "3"]
9
Using the provided matrix map of a city, where numbers represent travel time in minutes (all numbers are positive integers) and 'x' marks closed workshops, find the quickest route for Ben to travel from his current workshop at index (1, 8) to his destination workshop at index (4, 0), indexing from 0. Ben's car can move north, south, east, or west from a given crossroad, provided there's no x in that direction. Also, there are 3 districts in the city with district 1 covering rows 0 to 1, district 2 covering rows 2 to 4, and district 3 covering rows 5 to 8. Ben has to visit at least 1 workshop in each district on his path to the destination. The roads are bidirectional. The answer should be a list of tuples (in Python syntax) indicating the index of workshops on Ben's path. The start and end workshops must be included in the path. [6 9 19 x 5 9 x 15 x] [8 10 16 x x x 5 11 4] [5 11 20 1 14 16 7 1 5] [10 5 3 5 12 8 x x 2] [16 15 15 14 x 13 10 x 10] [6 4 7 13 x x x x 13] [x 1 6 3 x x 17 9 5] [x 11 x 9 x 5 7 x x] [x x x 18 x 2 x x 12]
traffic
pathfinding
1
[[1, 8], [2, 8], [2, 7], [2, 6], [2, 5], [2, 4], [2, 3], [3, 3], [3, 2], [3, 1], [4, 1], [5, 1], [5, 0], [4, 0]]
98
0.02410721778869629
14
4
4
[[["6", "9", "19", "x", "5", "9", "x", "15", "x"], ["8", "10", "16", "x", "x", "x", "5", "11", "4"], ["5", "11", "20", "1", "14", "16", "7", "1", "5"], ["10", "5", "3", "5", "12", "8", "x", "x", "2"], ["16", "15", "15", "14", "x", "13", "10", "x", "10"], ["6", "4", "7", "13", "x", "x", "x", "x", "13"], ["x", "1", "6", "3", "x", "x", "17", "9", "5"], ["x", "11", "x", "9", "x", "5", "7", "x", "x"], ["x", "x", "x", "18", "x", "2", "x", "x", "12"]]]
[[["6", "9", "19", "x", "5", "9", "x", "15", "x"], ["8", "10", "16", "x", "x", "x", "5", "11", "4"], ["5", "11", "20", "1", "14", "16", "7", "1", "5"], ["10", "5", "3", "5", "12", "8", "x", "x", "2"], ["16", "15", "15", "14", "x", "13", "10", "x", "10"], ["6", "4", "7", "13", "x", "x", "x", "x", "13"], ["x", "1", "6", "3", "x", "x", "17", "9", "5"], ["x", "11", "x", "9", "x", "5", "7", "x", "x"], ["x", "x", "x", "18", "x", "2", "x", "x", "12"]], [1, 8], [4, 0], 1, 4]
["[['6', '9', '19', 'x', '5', '9', 'x', '15', 'x'], ['8', '10', '16', 'x', 'x', 'x', '5', '11', '4'], ['5', '11', '20', '1', '14', '16', '7', '1', '5'], ['10', '5', '3', '5', '12', '8', 'x', 'x', '2'], ['16', '15', '15', '14', 'x', '13', '10', 'x', '10'], ['6', '4', '7', '13', 'x', 'x', 'x', 'x', '13'], ['x', '1', '6', '3', 'x', 'x', '17', '9', '5'], ['x', '11', 'x', '9', 'x', '5', '7', 'x', 'x'], ['x', 'x', 'x', '18', 'x', '2', 'x', 'x', '12']]", "(1, 8)", "(4, 0)", "1", "4"]
9
Alex is at a trampoline park with a grid of mini trampolines, arranged in a square of 9x9. Some trampolines are broken and unusable. A map of the park is provided below, with 1 indicating a broken trampoline and 0 indicating a functional one. Alex can jump to any of the eight adjacent trampolines, as long as they are not broken. However, Alex must make excatly 3 diagonal jumps, no more, no less, on his path to his destination. He is currently on the trampoline at position (1, 2) (positions are counted from 0, left to right, top to bottom) and wants to reach the trampoline at position (8, 8). What is the shortest sequence of trampolines he should jump on to reach his destination (including the first and final trampolines)? The answer should be a list of tuples, in Python syntax, indicating the row and column of each trampoline Alex jumps on. 1 0 1 1 1 0 1 1 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 0 1 1 0 1 0 1 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0
trampoline_matrix
pathfinding
9
[[1, 2], [2, 3], [3, 3], [4, 4], [5, 4], [6, 4], [7, 4], [8, 5], [8, 6], [8, 7], [8, 8]]
11
0.034554481506347656
11
8
2
["[[1, 0, 1, 1, 1, 0, 1, 1, 1], [0, 1, 0, 0, 1, 1, 1, 0, 0], [1, 0, 1, 0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0, 1, 1, 0], [1, 1, 0, 1, 0, 1, 0, 1, 1], [1, 0, 1, 1, 0, 1, 1, 1, 0], [0, 1, 0, 0, 0, 1, 1, 0, 0], [0, 1, 0, 0, 0, 1, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0, 0, 0]]", 3]
["[[1, 0, 1, 1, 1, 0, 1, 1, 1], [0, 1, 0, 0, 1, 1, 1, 0, 0], [1, 0, 1, 0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0, 1, 1, 0], [1, 1, 0, 1, 0, 1, 0, 1, 1], [1, 0, 1, 1, 0, 1, 1, 1, 0], [0, 1, 0, 0, 0, 1, 1, 0, 0], [0, 1, 0, 0, 0, 1, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0, 0, 0]]", [1, 2], [8, 8], 3]
["[[1, 0, 1, 1, 1, 0, 1, 1, 1], [0, 1, 0, 0, 1, 1, 1, 0, 0], [1, 0, 1, 0, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0, 1, 1, 0], [1, 1, 0, 1, 0, 1, 0, 1, 1], [1, 0, 1, 1, 0, 1, 1, 1, 0], [0, 1, 0, 0, 0, 1, 1, 0, 0], [0, 1, 0, 0, 0, 1, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0, 0, 0]]", "(1, 2)", "(8, 8)", "3"]
9
Given 5 labeled water jugs with capacities 129, 57, 45, 105, 128, 78 liters, we aim to fill 3 unlabeled buckets, numbered 1 to 3 and arranged in a line in ascending order, with 344, 376, 386 liters of water respectively. The amount of water in each unlabeled bucket can not at any point in time exceed the amount of water in the bucket placed after it. Jugs can only be filled to the top and emptied completely, and the unlabeled buckets cannot be overfilled. An action, represented as a tuple ('+', X, Y) or ('-', X, Y), involves adding to or removing water from the unlabeled bucket numbered Y, using the jug with capacity X. Determine the shortest sequence of actions needed to fill the buckets as specified, and present the solution as a list of action tuples in Python syntax.
water_jug
subset_sum
4
[["+", 128, 3], ["+", 129, 3], ["+", 129, 3], ["+", 105, 2], ["+", 105, 2], ["+", 128, 2], ["-", 45, 2], ["-", 45, 2], ["+", 128, 2], ["+", 78, 1], ["+", 105, 1], ["+", 128, 1], ["-", 45, 1], ["+", 78, 1]]
14
0.05803990364074707
14
36
3
[[129, 57, 45, 105, 128, 78], [344, 376, 386]]
[[129, 57, 45, 105, 128, 78], [344, 376, 386]]
["[129, 57, 45, 105, 128, 78]", "[344, 376, 386]"]
10
In the 8-puzzle game, you are given a grid with numbered square tiles arranged randomly and one tile missing. The goal is to arrange the tiles in descending order by sliding them into the empty space. The tiles can move in 4 directions: left, right, up, and down. Given the initial state of the puzzle below, where the empty spot is represented as “_”, provide the shortest list of tiles that need to be swapped with the empty spot to achieve the goal state. The goal state is when all tiles are in descending order, with the largest number in the top left corner, and the empty spot is in the bottom right corner. The solution should be a list of numbers in Python format, where each number represents the number on the tile that the empty spot is swapped with at each turn. Initial state of the puzzle: [[93, 88, 60], [28, 18, 24], ['_', 47, 53]]
8_puzzle
puzzle
3
[47, 53, 24, 18, 28, 47, 53, 24, 18, 28, 47, 53, 24, 18]
14
0.022923946380615234
14
4
9
[[[93, 88, 60], [28, 18, 24], ["_", 47, 53]]]
[[[93, 88, 60], [28, 18, 24], ["_", 47, 53]]]
["[[93, 88, 60], [28, 18, 24], ['_', 47, 53]]"]