meetforever/GRPO_CL huggingface dataset:
user: Given an integer array, return the k-th smallest distance among all the pairs. The distance of a pair (A, B) is defined as the absolute difference between A and B. 

Example 1:

Input:
nums = [1,3,1]
k = 1
Output: 0 
Explanation:
Here are all the pairs:
(1,3) -> 2
(1,1) -> 0
(3,1) -> 2
Then the 1st smallest distance pair is (1,1), and its distance is 0.



Note:

2 .
0 .
1 .

Answer: {"fn_name": "smallestDistancePair", "inputs": [[[1, 1, 3], 1]], "outputs": [0]}
user: You have an initial power P, an initial score of 0 points, and a bag of tokens.
Each token can be used at most once, has a value token[i], and has potentially two ways to use it.

If we have at least token[i] power, we may play the token face up, losing token[i] power, and gaining 1 point.
If we have at least 1 point, we may play the token face down, gaining token[i] power, and losing 1 point.

Return the largest number of points we can have after playing any number of tokens.
 



Example 1:
Input: tokens = [100], P = 50
Output: 0


Example 2:
Input: tokens = [100,200], P = 150
Output: 1


Example 3:
Input: tokens = [100,200,300,400], P = 200
Output: 2

 
Note:

tokens.length <= 1000
0 <= tokens[i] < 10000
0 <= P < 10000

Answer: {"fn_name": "bagOfTokensScore", "inputs": [[[100], 50]], "outputs": [0]}
user: Given an array A of integers, for each integer A[i] we need to choose either x = -K or x = K, and add x to A[i] (only once).
After this process, we have some array B.
Return the smallest possible difference between the maximum value of B and the minimum value of B.
 



Example 1:
Input: A = [1], K = 0
Output: 0
Explanation: B = [1]


Example 2:
Input: A = [0,10], K = 2
Output: 6
Explanation: B = [2,8]


Example 3:
Input: A = [1,3,6], K = 3
Output: 3
Explanation: B = [4,6,3]

 
Note:

1 <= A.length <= 10000
0 <= A[i] <= 10000
0 <= K <= 10000

Answer: {"fn_name": "smallestRangeII", "inputs": [[[1], 0]], "outputs": [0]}
user: Given 2n balls of k distinct colors. You will be given an integer array balls of size k where balls[i] is the number of balls of color i. 
All the balls will be shuffled uniformly at random, then we will distribute the first n balls to the first box and the remaining n balls to the other box (Please read the explanation of the second example carefully).
Please note that the two boxes are considered different. For example, if we have two balls of colors a and b, and two boxes [] and (), then the distribution [a] (b) is considered different than the distribution [b] (a) (Please read the explanation of the first example carefully).
We want to calculate the probability that the two boxes have the same number of distinct balls.
 
Example 1:
Input: balls = [1,1]
Output: 1.00000
Explanation: Only 2 ways to divide the balls equally:
- A ball of color 1 to box 1 and a ball of color 2 to box 2
- A ball of color 2 to box 1 and a ball of color 1 to box 2
In both ways, the number of distinct colors in each box is equal. The probability is 2/2 = 1

Example 2:
Input: balls = [2,1,1]
Output: 0.66667
Explanation: We have the set of balls [1, 1, 2, 3]
This set of balls will be shuffled randomly and we may have one of the 12 distinct shuffles with equale probability (i.e. 1/12):
[1,1 / 2,3], [1,1 / 3,2], [1,2 / 1,3], [1,2 / 3,1], [1,3 / 1,2], [1,3 / 2,1], [2,1 / 1,3], [2,1 / 3,1], [2,3 / 1,1], [3,1 / 1,2], [3,1 / 2,1], [3,2 / 1,1]
After that we add the first two balls to the first box and the second two balls to the second box.
We can see that 8 of these 12 possible random distributions have the same number of distinct colors of balls in each box.
Probability is 8/12 = 0.66667

Example 3:
Input: balls = [1,2,1,2]
Output: 0.60000
Explanation: The set of balls is [1, 2, 2, 3, 4, 4]. It is hard to display all the 180 possible random shuffles of this set but it is easy to check that 108 of them will have the same number of distinct colors in each box.
Probability = 108 / 180 = 0.6

Example 4:
Input: balls = [3,2,1]
Output: 0.30000
Explanation: The set of balls is [1, 1, 1, 2, 2, 3]. It is hard to display all the 60 possible random shuffles of this set but it is easy to check that 18 of them will have the same number of distinct colors in each box.
Probability = 18 / 60 = 0.3

Example 5:
Input: balls = [6,6,6,6,6,6]
Output: 0.90327

 
Constraints:

1 <= balls.length <= 8
1 <= balls[i] <= 6
sum(balls) is even.
Answers within 10^-5 of the actual value will be accepted as correct.

Answer: {"fn_name": "getProbability", "inputs": [[[1, 1]]], "outputs": [1.0]}
user: Suppose you have N integers from 1 to N. We define a beautiful arrangement as an array that is constructed by these N numbers successfully if one of the following is true for the ith position (1 
The number at the ith position is divisible by i.
i is divisible by the number at the ith position.




Now given N, how many beautiful arrangements can you construct?


Example 1:

Input: 2
Output: 2
Explanation: 
The first beautiful arrangement is [1, 2]:
Number at the 1st position (i=1) is 1, and 1 is divisible by i (i=1).
Number at the 2nd position (i=2) is 2, and 2 is divisible by i (i=2).
The second beautiful arrangement is [2, 1]:
Number at the 1st position (i=1) is 2, and 2 is divisible by i (i=1).
Number at the 2nd position (i=2) is 1, and i (i=2) is divisible by 1.



Note:

N is a positive integer and will not exceed 15.

Answer: {"fn_name": "countArrangement", "inputs": [[2]], "outputs": [2]}
user: There is a room with n lights which are turned on initially and 4 buttons on the wall. After performing exactly m unknown operations towards buttons, you need to return how many different kinds of status of the n lights could be.



Suppose n lights are labeled as number [1, 2, 3 ..., n], function of these 4 buttons are given below:


Flip all the lights.
Flip lights with even numbers.
Flip lights with odd numbers.
Flip lights with (3k + 1) numbers, k = 0, 1, 2, ...




Example 1:

Input: n = 1, m = 1.
Output: 2
Explanation: Status can be: [on], [off]




Example 2:

Input: n = 2, m = 1.
Output: 3
Explanation: Status can be: [on, off], [off, on], [off, off]




Example 3:

Input: n = 3, m = 1.
Output: 4
Explanation: Status can be: [off, on, off], [on, off, on], [off, off, off], [off, on, on].



Note:
n and m both fit in range [0, 1000].

Answer: {"fn_name": "flipLights", "inputs": [[1, 1]], "outputs": [2]}
user: Given an array A of non-negative integers, return the maximum sum of elements in two non-overlapping (contiguous) subarrays, which have lengths L and M.  (For clarification, the L-length subarray could occur before or after the M-length subarray.)
Formally, return the largest V for which V = (A[i] + A[i+1] + ... + A[i+L-1]) + (A[j] + A[j+1] + ... + A[j+M-1]) and either:

0 <= i < i + L - 1 < j < j + M - 1 < A.length, or
0 <= j < j + M - 1 < i < i + L - 1 < A.length.

 



Example 1:
Input: A = [0,6,5,2,2,5,1,9,4], L = 1, M = 2
Output: 20
Explanation: One choice of subarrays is [9] with length 1, and [6,5] with length 2.


Example 2:
Input: A = [3,8,1,3,2,1,8,9,0], L = 3, M = 2
Output: 29
Explanation: One choice of subarrays is [3,8,1] with length 3, and [8,9] with length 2.


Example 3:
Input: A = [2,1,5,6,0,9,5,0,3,8], L = 4, M = 3
Output: 31
Explanation: One choice of subarrays is [5,6,0,9] with length 4, and [3,8] with length 3.

 
Note:

L >= 1
M >= 1
L + M <= A.length <= 1000
0 <= A[i] <= 1000

Answer: {"fn_name": "maxSumTwoNoOverlap", "inputs": [[[0, 6, 5, 2, 2, 5, 1, 9, 4], 1, 2]], "outputs": [20]}
user: Given a wooden stick of length n units. The stick is labelled from 0 to n. For example, a stick of length 6 is labelled as follows:

Given an integer array cuts where cuts[i] denotes a position you should perform a cut at.
You should perform the cuts in order, you can change the order of the cuts as you wish.
The cost of one cut is the length of the stick to be cut, the total cost is the sum of costs of all cuts. When you cut a stick, it will be split into two smaller sticks (i.e. the sum of their lengths is the length of the stick before the cut). Please refer to the first example for a better explanation.
Return the minimum total cost of the cuts.
 
Example 1:

Input: n = 7, cuts = [1,3,4,5]
Output: 16
Explanation: Using cuts order = [1, 3, 4, 5] as in the input leads to the following scenario:

The first cut is done to a rod of length 7 so the cost is 7. The second cut is done to a rod of length 6 (i.e. the second part of the first cut), the third is done to a rod of length 4 and the last cut is to a rod of length 3. The total cost is 7 + 6 + 4 + 3 = 20.
Rearranging the cuts to be [3, 5, 1, 4] for example will lead to a scenario with total cost = 16 (as shown in the example photo 7 + 4 + 3 + 2 = 16).
Example 2:
Input: n = 9, cuts = [5,6,1,4,2]
Output: 22
Explanation: If you try the given cuts ordering the cost will be 25.
There are much ordering with total cost <= 25, for example, the order [4, 6, 5, 2, 1] has total cost = 22 which is the minimum possible.

 
Constraints:

2 <= n <= 10^6
1 <= cuts.length <= min(n - 1, 100)
1 <= cuts[i] <= n - 1
All the integers in cuts array are distinct.

Answer: {"fn_name": "minCost", "inputs": [[7, [1, 3, 4, 5, 0, 7]]], "outputs": [16]}
user: Given an array of integers arr. Return the number of sub-arrays with odd sum.
As the answer may grow large, the answer must be computed modulo 10^9 + 7.
 
Example 1:
Input: arr = [1,3,5]
Output: 4
Explanation: All sub-arrays are [[1],[1,3],[1,3,5],[3],[3,5],[5]]
All sub-arrays sum are [1,4,9,3,8,5].
Odd sums are [1,9,3,5] so the answer is 4.

Example 2:
Input: arr = [2,4,6]
Output: 0
Explanation: All sub-arrays are [[2],[2,4],[2,4,6],[4],[4,6],[6]]
All sub-arrays sum are [2,6,12,4,10,6].
All sub-arrays have even sum and the answer is 0.

Example 3:
Input: arr = [1,2,3,4,5,6,7]
Output: 16

Example 4:
Input: arr = [100,100,99,99]
Output: 4

Example 5:
Input: arr = [7]
Output: 1

 
Constraints:

1 <= arr.length <= 10^5
1 <= arr[i] <= 100

Answer: {"fn_name": "numOfSubarrays", "inputs": [[[1, 3, 5]]], "outputs": [4]}
user: Given two arrays of integers with equal lengths, return the maximum value of:
|arr1[i] - arr1[j]| + |arr2[i] - arr2[j]| + |i - j|
where the maximum is taken over all 0 <= i, j < arr1.length.
 
Example 1:
Input: arr1 = [1,2,3,4], arr2 = [-1,4,5,6]
Output: 13

Example 2:
Input: arr1 = [1,-2,-5,0,10], arr2 = [0,-2,-1,-7,-4]
Output: 20

 
Constraints:

2 <= arr1.length == arr2.length <= 40000
-10^6 <= arr1[i], arr2[i] <= 10^6

Answer: {"fn_name": "maxAbsValExpr", "inputs": [[[1, 2, 3, 4], [-1, 4, 5, 6]]], "outputs": [13]}
user: Given a char array representing tasks CPU need to do. It contains capital letters A to Z where different letters represent different tasks.Tasks could be done without original order. Each task could be done in one interval. For each interval, CPU could finish one task or just be idle.

However, there is a non-negative cooling interval n that means between two same tasks, there must be at least n intervals that CPU are doing different tasks or just be idle. 

You need to return the least number of intervals the CPU will take to finish all the given tasks.

Example 1:

Input: tasks = ["A","A","A","B","B","B"], n = 2
Output: 8
Explanation: A -> B -> idle -> A -> B -> idle -> A -> B.



Note:

The number of tasks is in the range [1, 10000].
The integer n is in the range [0, 100].

Answer: {"fn_name": "leastInterval", "inputs": [[["\"A\"", "\"A\"", "\"A\"", "\"B\"", "\"B\"", "\"B\""], 2]], "outputs": [8]}
user: The n-queens puzzle is the problem of placing n queens on an n×n chessboard such that no two queens attack each other.



Given an integer n, return the number of distinct solutions to the n-queens puzzle.

Example:


Input: 4
Output: 2
Explanation: There are two distinct solutions to the 4-queens puzzle as shown below.
[
 [".Q..",  // Solution 1
  "...Q",
  "Q...",
  "..Q."],

 ["..Q.",  // Solution 2
  "Q...",
  "...Q",
  ".Q.."]
]

Answer: {"fn_name": "totalNQueens", "inputs": [[4]], "outputs": [2]}
user: You are given an array x of n positive numbers. You start at point (0,0) and moves x[0] metres to the north, then x[1] metres to the west,
    x[2] metres to the south,
    x[3] metres to the east and so on. In other words, after each move your direction changes
    counter-clockwise.


    Write a one-pass algorithm with O(1) extra space to determine, if your path crosses itself, or not.



Example 1:

Given x = [2, 1, 1, 2],
?????
?   ?
???????>
    ?

Return true (self crossing)




Example 2:

Given x = [1, 2, 3, 4],
????????
?      ?
?
?
?????????????>

Return false (not self crossing)




Example 3:

Given x = [1, 1, 1, 1],
?????
?   ?
?????>

Return true (self crossing)



Credits:Special thanks to @dietpepsi for adding this problem and creating all test cases.

Answer: {"fn_name": "isSelfCrossing", "inputs": [[[2, 1, 1, 2]]], "outputs": [true]}
user: Suppose an array sorted in ascending order is rotated at some pivot unknown to you beforehand.

(i.e.,  [0,1,2,4,5,6,7] might become  [4,5,6,7,0,1,2]).

Find the minimum element.

The array may contain duplicates.

Example 1:


Input: [1,3,5]
Output: 1

Example 2:


Input: [2,2,2,0,1]
Output: 0

Note:


       This is a follow up problem to Find Minimum in Rotated Sorted Array.
       Would allow duplicates affect the run-time complexity? How and why?

Answer: {"fn_name": "findMin", "inputs": [[[1, 3, 5]]], "outputs": [1]}
user: You have n  tiles, where each tile has one letter tiles[i] printed on it.
Return the number of possible non-empty sequences of letters you can make using the letters printed on those tiles.
 
Example 1:
Input: tiles = "AAB"
Output: 8
Explanation: The possible sequences are "A", "B", "AA", "AB", "BA", "AAB", "ABA", "BAA".

Example 2:
Input: tiles = "AAABBC"
Output: 188

Example 3:
Input: tiles = "V"
Output: 1

 
Constraints:

1 <= tiles.length <= 7
tiles consists of uppercase English letters.

Answer: {"fn_name": "numTilePossibilities", "inputs": [["\"AAB\""]], "outputs": [89]}
user: Given two non-negative integers num1 and num2 represented as strings, return the product of num1 and num2, also represented as a string.

Example 1:


Input: num1 = "2", num2 = "3"
Output: "6"

Example 2:


Input: num1 = "123", num2 = "456"
Output: "56088"


Note:


       The length of both num1 and num2 is < 110.
       Both num1 and num2 contain only digits 0-9.
       Both num1 and num2 do not contain any leading zero, except the number 0 itself.
       You must not use any built-in BigInteger library or convert the inputs to integer directly.

Answer: {"fn_name": "multiply", "inputs": [["\"2\"", "\"3\""]], "outputs": ["600"]}
user: Given a m x n grid. Each cell of the grid has a sign pointing to the next cell you should visit if you are currently in this cell. The sign of grid[i][j] can be:

1 which means go to the cell to the right. (i.e go from grid[i][j] to grid[i][j + 1])
2 which means go to the cell to the left. (i.e go from grid[i][j] to grid[i][j - 1])
3 which means go to the lower cell. (i.e go from grid[i][j] to grid[i + 1][j])
4 which means go to the upper cell. (i.e go from grid[i][j] to grid[i - 1][j])

Notice that there could be some invalid signs on the cells of the grid which points outside the grid.
You will initially start at the upper left cell (0,0). A valid path in the grid is a path which starts from the upper left cell (0,0) and ends at the bottom-right cell (m - 1, n - 1) following the signs on the grid. The valid path doesn't have to be the shortest.
You can modify the sign on a cell with cost = 1. You can modify the sign on a cell one time only.
Return the minimum cost to make the grid have at least one valid path.
 
Example 1:

Input: grid = [[1,1,1,1],[2,2,2,2],[1,1,1,1],[2,2,2,2]]
Output: 3
Explanation: You will start at point (0, 0).
The path to (3, 3) is as follows. (0, 0) --> (0, 1) --> (0, 2) --> (0, 3) change the arrow to down with cost = 1 --> (1, 3) --> (1, 2) --> (1, 1) --> (1, 0) change the arrow to down with cost = 1 --> (2, 0) --> (2, 1) --> (2, 2) --> (2, 3) change the arrow to down with cost = 1 --> (3, 3)
The total cost = 3.

Example 2:

Input: grid = [[1,1,3],[3,2,2],[1,1,4]]
Output: 0
Explanation: You can follow the path from (0, 0) to (2, 2).

Example 3:

Input: grid = [[1,2],[4,3]]
Output: 1

Example 4:
Input: grid = [[2,2,2],[2,2,2]]
Output: 3

Example 5:
Input: grid = [[4]]
Output: 0

 
Constraints:

m == grid.length
n == grid[i].length
1 <= m, n <= 100

Answer: {"fn_name": "minCost", "inputs": [[[[1, 1, 1, 1], [2, 2, 2, 2], [1, 1, 1, 1], [2, 2, 2, 2], [], []]]], "outputs": [-1]}
user: Given a single positive integer x, we will write an expression of the form x (op1) x (op2) x (op3) x ... where each operator op1, op2, etc. is either addition, subtraction, multiplication, or division (+, -, *, or /).  For example, with x = 3, we might write 3 * 3 / 3 + 3 - 3 which is a value of 3.
When writing such an expression, we adhere to the following conventions:

The division operator (/) returns rational numbers.
There are no parentheses placed anywhere.
We use the usual order of operations: multiplication and division happens before addition and subtraction.
It's not allowed to use the unary negation operator (-).  For example, "x - x" is a valid expression as it only uses subtraction, but "-x + x" is not because it uses negation.

We would like to write an expression with the least number of operators such that the expression equals the given target.  Return the least number of operators used.
 

Example 1:
Input: x = 3, target = 19
Output: 5
Explanation: 3 * 3 + 3 * 3 + 3 / 3.  The expression contains 5 operations.

Example 2:

Input: x = 5, target = 501
Output: 8
Explanation: 5 * 5 * 5 * 5 - 5 * 5 * 5 + 5 / 5.  The expression contains 8 operations.


Example 3:
Input: x = 100, target = 100000000
Output: 3
Explanation: 100 * 100 * 100 * 100.  The expression contains 3 operations.
 



Note:

2 <= x <= 100
1 <= target <= 2 * 10^8

Answer: {"fn_name": "leastOpsExpressTarget", "inputs": [[3, 19]], "outputs": [5]}
user: We write the integers of A and B (in the order they are given) on two separate horizontal lines.
Now, we may draw connecting lines: a straight line connecting two numbers A[i] and B[j] such that:

A[i] == B[j];
The line we draw does not intersect any other connecting (non-horizontal) line.

Note that a connecting lines cannot intersect even at the endpoints: each number can only belong to one connecting line.
Return the maximum number of connecting lines we can draw in this way.
 
Example 1:

Input: A = [1,4,2], B = [1,2,4]
Output: 2
Explanation: We can draw 2 uncrossed lines as in the diagram.
We cannot draw 3 uncrossed lines, because the line from A[1]=4 to B[2]=4 will intersect the line from A[2]=2 to B[1]=2.


Example 2:
Input: A = [2,5,1,2,5], B = [10,5,2,1,5,2]
Output: 3


Example 3:
Input: A = [1,3,7,1,7,5], B = [1,9,2,5,1]
Output: 2
 


Note:

1 <= A.length <= 500
1 <= B.length <= 500
1 <= A[i], B[i] <= 2000

Answer: {"fn_name": "maxUncrossedLines", "inputs": [[[1, 4, 2], [1, 2, 4]]], "outputs": [2]}
user: Given the coordinates of four points in 2D space, return whether the four points could construct a square.

The coordinate (x,y) of a point is represented by an integer array with two integers.

Example:

Input: p1 = [0,0], p2 = [1,1], p3 = [1,0], p4 = [0,1]
Output: True



 Note: 

All the input integers are in the range [-10000, 10000].
A valid square has four equal sides with positive length and four equal angles (90-degree angles).
Input points have no order.

Answer: {"fn_name": "validSquare", "inputs": [[[0, 0], [1, 1], [1, 0], [0, 1]]], "outputs": [true]}
user: Given an integer array arr, you should partition the array into (contiguous) subarrays of length at most k. After partitioning, each subarray has their values changed to become the maximum value of that subarray.
Return the largest sum of the given array after partitioning.
 
Example 1:
Input: arr = [1,15,7,9,2,5,10], k = 3
Output: 84
Explanation: arr becomes [15,15,15,9,10,10,10]

Example 2:
Input: arr = [1,4,1,5,7,3,6,1,9,9,3], k = 4
Output: 83

Example 3:
Input: arr = [1], k = 1
Output: 1

 
Constraints:

1 <= arr.length <= 500
0 <= arr[i] <= 109
1 <= k <= arr.length

Answer: {"fn_name": "maxSumAfterPartitioning", "inputs": [[[1, 15, 7, 9, 2, 5, 10], 3]], "outputs": [84]}
user: Some people will make friend requests. The list of their ages is given and ages[i] is the age of the ith person. 
Person A will NOT friend request person B (B != A) if any of the following conditions are true:

age[B] <= 0.5 * age[A] + 7
age[B] > age[A]
age[B] > 100 && age[A] < 100

Otherwise, A will friend request B.
Note that if A requests B, B does not necessarily request A.  Also, people will not friend request themselves.
How many total friend requests are made?
Example 1:
Input: [16,16]
Output: 2
Explanation: 2 people friend request each other.

Example 2:
Input: [16,17,18]
Output: 2
Explanation: Friend requests are made 17 -> 16, 18 -> 17.
Example 3:
Input: [20,30,100,110,120]
Output: 3
Explanation: Friend requests are made 110 -> 100, 120 -> 110, 120 -> 100.

 
Notes:

1 <= ages.length <= 20000.
1 <= ages[i] <= 120.

Answer: {"fn_name": "numFriendRequests", "inputs": [[[16, 16]]], "outputs": [2]}
user: Return the number of distinct non-empty substrings of text that can be written as the concatenation of some string with itself (i.e. it can be written as a + a where a is some string).
 
Example 1:
Input: text = "abcabcabc"
Output: 3
Explanation: The 3 substrings are "abcabc", "bcabca" and "cabcab".

Example 2:
Input: text = "leetcodeleetcode"
Output: 2
Explanation: The 2 substrings are "ee" and "leetcodeleetcode".

 
Constraints:

1 <= text.length <= 2000
text has only lowercase English letters.

Answer: {"fn_name": "distinctEchoSubstrings", "inputs": [["\"abcabcabc\""]], "outputs": [3]}
user: Given an integer array with all positive numbers and no duplicates, find the number of possible combinations that add up to a positive integer target.

Example:

nums = [1, 2, 3]
target = 4

The possible combination ways are:
(1, 1, 1, 1)
(1, 1, 2)
(1, 2, 1)
(1, 3)
(2, 1, 1)
(2, 2)
(3, 1)

Note that different sequences are counted as different combinations.

Therefore the output is 7.



Follow up:
What if negative numbers are allowed in the given array?
How does it change the problem?
What limitation we need to add to the question to allow negative numbers? 

Credits:Special thanks to @pbrother for adding this problem and creating all test cases.

Answer: {"fn_name": "combinationSum4", "inputs": [[[1, 2, 3], 4]], "outputs": [7]}
user: There are two types of soup: type A and type B. Initially we have N ml of each type of soup. There are four kinds of operations:

Serve 100 ml of soup A and 0 ml of soup B
Serve 75 ml of soup A and 25 ml of soup B
Serve 50 ml of soup A and 50 ml of soup B
Serve 25 ml of soup A and 75 ml of soup B

When we serve some soup, we give it to someone and we no longer have it.  Each turn, we will choose from the four operations with equal probability 0.25. If the remaining volume of soup is not enough to complete the operation, we will serve as much as we can.  We stop once we no longer have some quantity of both types of soup.
Note that we do not have the operation where all 100 ml's of soup B are used first.  
Return the probability that soup A will be empty first, plus half the probability that A and B become empty at the same time.
 
Example:
Input: N = 50
Output: 0.625
Explanation: 
If we choose the first two operations, A will become empty first. For the third operation, A and B will become empty at the same time. For the fourth operation, B will become empty first. So the total probability of A becoming empty first plus half the probability that A and B become empty at the same time, is 0.25 * (1 + 1 + 0.5 + 0) = 0.625.


Notes: 

0 <= N <= 10^9. 
Answers within 10^-6 of the true value will be accepted as correct.

Answer: {"fn_name": "soupServings", "inputs": [[50]], "outputs": [0.625]}