Ckriman/General365_Public huggingface dataset:
Question: A solid 4×4×4 cube is built from unit cubes (each with side length 1). In each of the three orthographic views (front, top, and left), we label the 4×4 grid by coordinates (i, j), where i is the column number from left to right and j is the row number from bottom to top; thus the cell in the 1st column and 2nd row (counting from the lower-left corner) is (1,2).

We remove some unit cubes to create through-holes in the cube. A listed through-hole location in a given view means that all unit cubes along the line of sight at that (i, j) position are removed, so the hole passes completely through the cube.

Under the following constraints, what is the maximum possible remaining volume of the cube?

1. Front view: (3,2) and (4,3) are through-hole locations.
2. Left view: (2,4), (3,3), and (4,2) are through-hole locations.
3. Top view: (3,3) and (4,2) are through-hole locations.
Answer: \boxed{41}
Question: A solid cube of side length 5 is built from 1×1×1 unit cubes (a 5×5×5 cube).

In the front, top, and left orthographic views, each face is represented as a 5×5 grid. Coordinates are 1-indexed, with (1,2) denoting the grid cell in the first column from the left and the second row from the bottom.

By removing some unit cubes, you create straight through-holes (tunnels) that pass completely through the cube. In a given view, a listed cell indicates a tunnel parallel to that view's line of sight (front view: front-to-back; left view: left-to-right; top view: top-to-bottom). The through-holes must satisfy:

1. Front view: the cells (3,3) and (4,2) are through-holes.
2. Left view: the cells (2,3) and (4,4) are through-holes.
3. Top view: the cells (3,4) and (4,3) are through-holes.

What is the maximum possible remaining volume of the cube?
Answer: \boxed{98}
Question: A solid 4×4×4 cube is built from unit cubes (each with side length 1). In each of the three orthographic views (front, top, and left), we label the 4×4 grid by coordinates (i, j), where i is the column number from left to right and j is the row number from bottom to top; thus the cell in the 1st column and 2nd row (counting from the lower-left corner) is (1, 2).

By removing some unit cubes, we create through-holes through the cube. A listed through-hole location in a given view means that all unit cubes along the line perpendicular to that face at that (i, j) position are removed, so the hole passes completely through the cube.

Under the following constraints, what is the maximum possible remaining volume of the cube?

1. Front view: (2, 3) and (3, 2) are through-hole locations.
2. Left view: (2, 2), (3, 3), and (4, 4) are through-hole locations.
3. Top view: (2, 2) and (3, 4) are through-hole locations.
Answer: \boxed{40}
Question: A party game uses a blank 4×4 grid on a round table, which must be filled with 16 different cards, each labeled with an actor's name. Everyone seated around the table takes turns clockwise. On each turn, a player must choose exactly one of the following three moves:

[Randomly draw one actor card and place it in an empty cell of the grid.]
[Swap the positions of any two actor cards already placed on the grid.]
[Move one actor card already on the grid to an empty cell, leaving its original cell empty.]

The draw pile initially contains 16 cards, and once a card is drawn, it cannot be returned to the pile. The game ends immediately when the last card is drawn and placed.

Tonight, you host the game and invite one friend (two players total). After you both sit down, you may choose who goes first. When preparing the 16 actor cards for the draw pile, what is the minimum number of cards of actors you like that must be included to guarantee that, when the game ends, the first row (the top row) of the grid contains at least one actor you like?
Answer: \boxed{2}
Question: A party game uses a 5×5 empty grid that must be filled with 25 different director-name cards. Everyone seated around the table takes turns clockwise to take an "action". On each turn, a player must choose exactly one of the following three actions:

1. Draw one director card at random and place it in an empty cell of the grid.

2. Swap the positions of any two cards already on the grid.

3. Move one card already on the grid to an empty cell, leaving its original cell empty.

The deck starts with exactly 25 cards, and drawn cards are not returned. The game ends when the last card is drawn and placed.

Tonight, as the host, you invite one friend to play at your home. After you both sit down, you get to decide who goes first. When preparing the 25 director cards for this game, what is the minimum number of cards featuring your favorite directors that must be included to guarantee that, when the game ends, the bottom row of the grid contains at least one of your favorite directors?
Answer: \boxed{2}
Question: In a party game, there is a blank 5×4 grid on a round table that must be filled with 20 different cards labeled with scientists' names. All players seated around the table take turns clockwise to make a move. On each turn, a player must choose exactly one of the following three actions:

1. Randomly draw one scientist card and place it in an empty cell of the grid.
2. Swap the positions of any two scientist cards already placed on the grid.
3. Move one scientist card already on the grid to an empty cell, leaving its original cell empty.

The draw pile initially contains 20 cards, and once a card is drawn, it cannot be returned to the pile. The game ends immediately when the last card is drawn and placed.

Tonight, you host the game and invite one friend (two players total). After you both sit down, you may choose who goes first. When preparing the 20 scientist cards for the draw pile, what is the minimum number of cards bearing the names of scientists you like that must be included to guarantee that, when the game ends, the first column (counting from left to right) of the grid contains at least one scientist you like?
Answer: \boxed{2}
Question: The core code of a new computer virus, V, is unknown. All that is known is that its code ID is a non-negative integer less than 800, and it is equally likely to be odd or even (50% each). If virus V swaps its core code with that of an antivirus program, their roles (destructive vs. protective) are swapped as well.

It is known that V has a destructive mode: once triggered, it can format all hard drives in under one second. In this mode, if V's core code ID is changed to the antivirus program's code ID, it behaves like a harmless no-op (NOP). The antivirus program's core code ID is 120.

Now suppose virus V and the antivirus program swap core codes once per second, and this continues until, at the end of some swap, V's core code ID equals the total number of swaps that have occurred so far. A computer infected with this virus is left unused for several days and then powered on. What is the probability that its hard drives will not be formatted?
Answer: \boxed{0}
Question: An ancient incantation has an unknown stability coefficient. It is an integer less than 600, and it is equally likely to be odd or even (probability 1/2 each).

The "Silence Spell" has stability coefficient 90. If the stability coefficients of the incantation and the Silence Spell are swapped, then their magical effects are swapped as well (destructive vs. ineffective).

When the incantation is unstable, it triggers a "spacetime collapse" with an affected range exceeding 1 kilometer, and it forms in under 1 minute. However, if during this unstable state the incantation's stability coefficient is changed to match the Silence Spell's coefficient (90), the magic dissipates.

Now the incantation's stability coefficient and the Silence Spell's coefficient are swapped once every minute. This continues until, immediately after some swap, the incantation's stability coefficient equals the total number of swaps performed so far. A relic sealed with this incantation is left untouched for 100 years and is then touched.

What is the probability that touching it does not trigger a spacetime collapse?
Answer: \boxed{0}
Question: The ID of a synthetic virus's pathogenic gene sequence is unknown. All that is known is that this ID is a positive integer less than 400, and it is equally likely to be odd or even (50% each). If the virus swaps its gene sequence with that of a probiotic, their properties (lethal vs. beneficial) are swapped as well.

It is known that the virus is certainly fatal: after infection, no one survives more than one hour. In this setting, if the virus's gene-sequence ID is changed to the probiotic's gene-sequence ID, then it behaves like an ordinary probiotic. The probiotic's gene-sequence ID is 128.

Now suppose the virus and the probiotic swap gene sequences once every hour, and this continues until, at the end of some swap, the virus's gene-sequence ID equals the total number of swaps that have occurred so far. A sample of this virus is left in the lab for several weeks and then leaks. What is the probability that a person exposed to it survives?
Answer: \boxed{0}
Question: Each straight cut counts as one cut. Given a sufficiently large sheet of paper that can be folded, with negligible thickness, what is the minimum number of straight cuts needed to produce exactly 1999 rhombic holes, each with interior angles of 60° and 120°?
Answer: \boxed{1}
Question: Each straight cut counts as one cut. You have a sufficiently large sheet of paper that can be folded, with negligible thickness. What is the minimum number of straight cuts needed to produce exactly 99 rhombuses, each with a side length of 2 cm?
Answer: \boxed{1}
Question: Each straight cut counts as one cut. You have a sufficiently large sheet of paper that can be folded, with negligible thickness. What is the minimum number of straight cuts needed to produce exactly 365 square holes, each with an area of 9 cm²?
Answer: \boxed{1}
Question: Four expedition teams (Azure Dragon, White Tiger, Vermilion Bird, and Black Tortoise) each explored a different ruin (Desert Temple, Jungle Altar, Snow Mountain Fortress, Underwater Ancient City). Each team has a different number of members (ranging from 5 to 8), and they brought back 2, 3, or 4 artifacts. It is known that: 1. The Vermilion Bird team has 5 members. 2. The White Tiger team explored the Desert Temple. 3. The team with 6 members brought back 1 more artifact than the team that explored the Jungle Altar. 4. Either the Azure Dragon team or the team that explored the Snow Mountain Fortress brought back 3 artifacts. 5. The team that explored the Underwater Ancient City brought back 2 more artifacts than the Black Tortoise team. How many possible values are there for the total number of artifacts brought back by the four teams?
Answer: \boxed{2}
Question: Four children-Timmy, Rosie, Jack, and Lily-each keep a different pet: cat, dog, rabbit, or hamster. Each child is of a different age, ranging from 8 to 11 years old, and they feed their pet 1, 2, or 3 times per day. It is known that:
1. Jack is 8 years old.
2. Rosie keeps the cat.
3. The 9-year-old child feeds 1 more time per day than the child who keeps the dog.
4. Either Timmy or the child who keeps the rabbit feeds their pet 2 times per day.
5. The child who keeps the hamster feeds 2 more times per day than Lily.
Question: How many possible values are there for the total number of daily feeding times for all four children?
Answer: \boxed{2}
Question: Four colleagues-Kevin, Lisa, Mike, and Sophia-are each in charge of a different project: A, B, C, or D. Each person works on a different floor, ranging from 5 to 8, and last month, they completed 2, 3, and 4 tasks respectively. It is known that:
1. Mike works on the 5th floor.
2. Lisa is in charge of project A.
3. The person working on the 6th floor completed 1 more task than the person in charge of project B.
4. Between Kevin and the person in charge of Project C, one completed 3 tasks.
5. The person in charge of project D completed 2 more tasks than Sophia.
Question: How many possible values are there for the total number of tasks completed by the four colleagues?
Answer: \boxed{2}
Question: In this world, "+" denotes multiplication, "-" denotes addition, "×" denotes subtraction, and "÷" denotes division. Furthermore, the order of operations is reversed, but the positions of the dividend and divisor stay the same in division. For example, 2 ÷ 1 = 2 instead of 0.5.  
Question: 10 + 3 × 3 ÷ (2 + 2) = ?
Answer: \boxed{0}
Question: In this world, "+" denotes multiplication, "−" denotes addition, "×" denotes subtraction, and "÷" denotes division. Moreover, the order of precedence of all operations is also reversed; however, in division, the positional relationship between the dividend and the divisor remains unchanged. For example, (2 ÷ 1 = 2), rather than 0.5.

Question:
9 + 1 × 1 ÷ (2 + 6) = ?
Answer: \boxed{0}
Question: In this alternate world, the operations are redefined as follows: + denotes multiplication, - denotes addition, × times denotes subtraction, and ÷ denotes division. Furthermore, the standard order of operations (precedence) is completely reversed. However, the roles of the dividend and divisor in division remain the same (e.g., 2÷1=2, not 0.5).Question: What is the value of 5+5×4÷(5+2)=?
Answer: \boxed{0.5}
Question: A city needs to deliver supplies to three stations (Station 1, Station 2, and Station 3):
Demand: Station 1 requires 22 tons, Station 2 requires 4 tons, Station 3 requires 4 tons.
Vehicle load limit: 10 tons per trip.
Special constraint: Deliveries to Station 2 can only begin after deliveries to Station 1 are completed.
Transportation cost: The transportation cost is 2 yuan per kilometer.
Distances:
Warehouse to Station 1 = 10 km
Warehouse to Station 2 = 14 km
Warehouse to Station 3 = 18 km
Station 1 to Station 2 = 5 km
Station 1 to Station 3 = 15 km
Station 2 to Station 3 = 5 km
Vehicles must return to the warehouse at the end of their routes.
Question: What is the minimum transportation cost in yuan that satisfies all of the above requirements?
Answer: \boxed{156}
Question: An international aid organization needs to deliver rescue supplies to three regions (A1, B2, and C3):
Demand: Region A1 requires 4 tons, Region B2 requires 21 tons, Region C3 requires 5 tons.
Vehicle load limit: 10 tons per trip.
Special constraint: Deliveries to Region A1 can only begin after deliveries to Region B2 are completed.
Transportation cost: The transportation cost is 2.5 yuan per kilometer.
Distances:
Warehouse to A1 = 14 km
A1 to B2 = 5 km
B2 to C3 = 6 km
Warehouse to C3 = 16 km
Warehouse to B2 = 10 km
A1 to C3 = 7 km
Vehicles must return to the warehouse at the end of their routes.
Question: What is the minimum transportation cost in yuan that satisfies all of the above requirements?
Answer: \boxed{192.5}
Question: A supply chain company needs to deliver raw materials to three factories: upstream, midstream, and downstream.
Demand: The upstream factory requires 9 tons, the midstream factory requires 9 tons, and the downstream factory requires 12 tons.
Vehicle load limit: 15 tons per trip.
Special constraint: Deliveries to the midstream factory can only begin after deliveries to the upstream factory are completed.
Transportation cost: The transportation cost is 3 yuan per kilometer.
Distances:
Warehouse to upstream factory = 25 km
Upstream to midstream factory = 8 km
Midstream to downstream factory = 10 km
Warehouse to downstream factory = 30 km
Warehouse to midstream factory = 20 km
Upstream to downstream factory = 15 km
Vehicles must return to the warehouse at the end of their routes.
Question: What is the minimum transportation cost in yuan that satisfies all of the above requirements?
Answer: \boxed{339}
Question: A beverage stand is hosting a "Summer Refreshment" event: 3 empty cups can be exchanged for 1 new drink. A group of students plans to participate. Detailed Event Rules: Every 3 empty cups can be redeemed for 1 new drink. Each new drink consumed generates another empty cup. This process can be repeated as long as there are at least 3 empty cups. Each drink costs 3 yuan. All transactions must be completed on the same day. Drinks cannot be shared; each student must receive one full drink. The stand allows returns at the original price for items purchased on the same day, provided the total refund amount does not exceed 16 yuan.
Question: There are 45 students in the group. How many different methods are there to ensure every student gets a drink at the minimum total cost? (Methods are distinguished only by the total number of drinks purchased and the total number of drinks returned; combinations with the same quantities are considered identical.)
Answer: \boxed{2}
Question: The Delicious Yogurt Shop is running a "Green Cup Recycling" promotion: 4 empty yogurt cups can be exchanged for 1 new yogurt. A tour group is interested in this offer. Detailed Event Rules: Every 4 empty cups can be redeemed for 1 new yogurt. Each new yogurt consumed generates another empty cup. This process can be repeated as long as there are at least 4 empty cups. Each yogurt costs 5 yuan. All transactions must be completed on the same day. Yogurts cannot be shared; each person must receive one full cup. The shop allows returns at the original price for items purchased on the same day, provided the total refund amount does not exceed 15 yuan.
Question: There are 34 people in the tour group. How many different methods are there to ensure every person gets a yogurt at the minimum total cost? (Methods are distinguished only by the total number of yogurts purchased and the total number of yogurts returned; combinations with the same quantities are considered identical.)
Answer: \boxed{1}
Question: The Lohas Café has launched a "Coffee Cup Recycling Program": 5 empty coffee cups can be exchanged for 1 new cup of coffee. A company has decided to provide this as a benefit for its employees. Detailed Event Rules: Every 5 empty coffee cups can be redeemed for 1 new cup of coffee. Each new coffee consumed generates another empty cup. This process can be repeated as long as there are at least 5 empty cups. Each cup of coffee costs 6 yuan. All transactions must be completed on the same day. Coffee cannot be shared; each employee must receive one full cup. The café allows returns at the original price for items purchased on the same day, provided the total refund amount does not exceed 25 yuan.
Question: The company has 55 employees. How many different methods are there to ensure every employee gets a coffee at the minimum total cost? (Methods are distinguished only by the total number of coffees purchased and the total number of coffees returned; combinations with the same quantities are considered identical.)
Answer: \boxed{4}
Question: Define three new operators. The operators ▶ and ➕ are related to binary operations, while ▷ is essentially a comparison-based operation.
6▶6=1,39▶93=00,770▶707=100
1▷2=1,200▷199=199,505▷505=505
1➕2=110,2➕4=10100,3➕5=11101
Find the value of: 765 ▶ 756 ▷ (3 ➕ 5) =?
Answer: \boxed{100}
Question: Define three new operators, where ▶ and ➕ are related to binary representations, and ▷ is essentially a comparison-based operation.
3▶3=1,25▶25=11,89▶98=00
5▷9=5,10▷10=10,21▷12=12
1➕1=2,2➕2=20,3➕3=22
Find the value of: 101▶111▷(2➕3)=?
Answer: \boxed{21}
Question: Define three new operators, where ▶ and ➕ are related to binary representations, and ▷ is essentially a comparison-based operation.
3▶4=0,77▶77=11,818▶808=101
9▷19=9,201▷102=102,50▷50=50
1➕8=1001,2➕5=111,3➕6=121
Find the value of: (11➕12)▷(246▶256)=?
Answer: \boxed{101}
Question: A plastic tote has interior dimensions of 0.8 m (length) × 0.6 m (width) × 0.5 m (height). There is a notch on the top edge of one long side, extending 10 cm downward. Assuming no leakage from other parts and regardless of the tote's orientation, what is the maximum volume of liquid (cubic meters, m³) that the tote can hold?
Answer: \boxed{0.24}
Question: A custom glass water tank has interior dimensions of 2 m (length) × 1 m (width) × 1.5 m (height). There is a crack on the front panel, located 30 cm from one of the 2m edges. Assuming the water tank can be oriented in any direction on level ground and ignoring leakage elsewhere, what is the maximum volume of water (cubic meters, m³)) it can hold ?
Answer: \boxed{3}
Question: A large wooden cargo crate has interior dimensions of 3 m (length) × 2 m (width) × 2 m (height). There is a small opening located at the geometric center of one of the 2 m × 2 m side panels. Assuming the crate can be oriented in any direction on the ground and ignoring leakage elsewhere, what is the maximum volume of grain (cubic meters, m³) that the crate can hold?
Answer: \boxed{12}
Question: There are three containers with capacities of 1600 ml, 1000 ml, and 600 ml. With an unlimited supply of water, what is the minimum number of steps required to measure out exactly 800 ml of water? (Note: Only the act of transferring water between containers counts as one step.)
Answer: \boxed{3}
Question: There are three containers with capacities of 800 ml, 500 ml, and 300 ml. With an unlimited supply of water, what is the minimum number of steps required to obtain exactly 400 ml of water? (Note: Only the transfer of water between two containers counts as one step.)
Answer: \boxed{3}
Question: There are three containers with capacities of 2200 ml, 1200 ml, and 800 ml. With an unlimited supply of water, what is the minimum number of steps required to measure out exactly 1800 ml of water? (Note: Only the transfer of water between containers counts as one step.)
Answer: \boxed{2}
Question: Magician Alfred is performing a miraculous number transformation. He has eight magic hats numbered 1 through 8, where the number of crystals in each hat initially corresponds to its label. Alfred performs a series of "doubling spells": in each step, he chooses two hats, then pours crystals from one hat into another such that the number of crystals in the receiving hat is exactly doubled. Alfred aims to reach a final state where the number of crystals in hats 1 through 8 are 3, 4, 3, 5, 3, 7, 3, 8, respectively. Is this transformation possible? If so, provide a sequence of operations to achieve it.
Answer: It is impossible
Question: Magician Oscar is performing an impressive number-based magic act. He has nine magic hats numbered 1 through 9, where each hat initially contains a number of gold coins equal to its label. Oscar performs a series of "doubling spells": in each step, he chooses two hats and pours coins from one hat into another such that the number of coins in the receiving hat is exactly doubled.
Oscar aims to achieve a final configuration where the number of gold coins in hats 1 through 9 are 5, 3, 5, 6, 5, 7, 5, 4, 5, respectively. Is this transformation possible? If so, provide a sequence of operations to achieve it.
Answer: It is impossible.
Question: Magician Luna is performing a mysterious gem-based magic show. She has twelve magic hats numbered 1 through 12, where each hat initially contains a number of gems equal to its label. Luna can perform a series of "doubling spells": in each step, she chooses two hats and pours gems from one hat into another such that the number of gems in the receiving hat is exactly doubled.
Luna aims to achieve the final configuration (in order from hat 1 to hat 12): 7, 4, 7, 5, 7, 2, 7, 7, 7, 8, 7, 10. Is this transformation possible? If so, outline a sequence of operations that accomplishes this.
Answer: It is impossible.
Question: There is a periodic sequence {11, 21, 31, 41, 51, 61, 71, 81, ... } with a period length n, where n < 12. If, for every possible value of the number of terms N that satisfies the given conditions, all factors of N must belong to the set {m_{1}, m_{2}, m_{3}, ... }, what is the sum of the elements in this set?
Answer: \boxed{53}
Question: This is a periodic sequence {5, 10, 15, 20, 25, 30, ... } with a period length of n, where n < 10. If, for every possible value of the number of terms N that satisfies the given conditions, all factors of N must belong to the set {m_1, m_2, m_3, ... }, what is the sum of the elements in this set?
Answer: \boxed{40}
Question: There is a periodic sequence {15, 25, 35, 45, 55, 65, 75, ... } with a period length n, where n < 9. If, for every possible value of the number of terms N in the sequence that satisfies the conditions, all factors of N must belong to the set {m_1, m_2, m_3, ... }, what is the sum of the elements in this set?
Answer: \boxed{22}
Question: A fraud case occurred at Liam's house. At the time of the incident, six people were present: Liam's father, mother, older brother, maternal aunt (mother's sister), paternal uncle (father's brother), and Liam himself. The police investigation concluded the following:
1. The fraudster and the victim are not directly related by blood.
2. At the time of the incident, one male and one female were in the basement billiard room for recreation.
3. The crime took place in a library, where the fraudster and the victim met.
4. At the time of the incident, Liam's parents were in different locations.
5. There was exactly one fraudster and exactly one victim among the people in the house.
6. The two people in the billiard room were not directly related by blood.
7. The fraudster is younger than the victim.
8. Two people of the same gender were swimming in the pool at the time of the incident.
9. At the time of the incident, the mother's son and daughter were each alone.
10. The mother was not swimming at the time of the incident.
11. The father was not the victim.
Question: Where was the father's brother (paternal uncle) at the time of the incident?
Answer: In the billiard room.
Question: A serious poisoning incident occurred at the ancestral mansion of the century-old Jin Pharmaceuticals. At the time of the incident, six people were present: the head of the family (male), his wife (female), their eldest son, the wife's brother (maternal uncle), the head's sister (paternal aunt), and their eldest daughter.
The police investigation established the following facts:
1. The victim and the perpetrator are not direct blood relatives (i.e., not a parent-child relationship).
2. At the time of the incident, one male and one female were having a private conversation in the downstairs tea room.
3. The crime took place in the study, where the victim and the perpetrator had a heated confrontation.
4. The head of the family and his wife were not together when the incident occurred.
5. Among the six people in the house, there was exactly one perpetrator and exactly one victim.
6. The two individuals in the tea room are not direct blood relatives.
7. In terms of age, the perpetrator is younger than the victim.
8. Two people of the same gender were walking in the back garden at the time of the incident.
9. At the time of the incident, the wife's two children (the eldest son and eldest daughter) were in separate areas.
10. The wife was not in the back garden.
11. The head of the family was not the victim.
Question: Where was the head's sister (paternal aunt) at the time of the incident?
Answer: Study
Question: A threat incident occurred at Noah's home. At the time of the incident, six people were present: Noah's father, mother, younger sister, mother's sister (maternal aunt), father's brother (paternal uncle), and Noah himself. The police investigation concluded the following:
1. The threatener and the victim are not direct blood relatives.
2. At the time of the incident, one male and one female were in the sixth-floor yoga room.
3. The crime took place in a supermarket, where the threatener and the victim encountered each other.
4. At the time of the incident, Noah's parents were in different locations.
5. Among the people in the house, there was exactly one threatener and exactly one victim.
6. The two individuals in the yoga room were not direct blood relatives.
7. In terms of age, the threatener is older than the victim.
8. Two people of the same gender were watering plants on the balcony at the time of the incident.
9. At the time of the incident, the mother's son and daughter (Noah and his sister) were each engaged in separate activities.
10. The mother was not watering plants at the time of the incident.
11. The father was not the victim.
Question: Where was the father's brother (paternal uncle) at the time of the incident?
Answer: In the yoga room.
Question: Six individuals-A, B, C, D, E, and F-attend a 3D movie. The theater has 12 seats arranged in four rows, with three seats per row. No one is seated in front of A. B is seated in the row immediately in front of A. D and A are seated in the same row. C and E occupy adjacent seats (left-right neighbors in the same row). A is seated in front of C. F is seated far away from A. Both B and F state that they can see the exact center of the screen. F tells B, "But my view is clearer than yours (meaning: I am seated further back)." D says, "Fortunately, no one is blocking my view of the screen." 
Row 1 is the row closest to the screen. The center seat of any row (the middle column) provides a direct view of the exact center of the screen. Let Z_{11} represents the leftmost seat in the first row, and Z_{43} represents the third seat in the fourth row. Question: Among these six people, what fraction have positions that are uniquely determined?
Answer: \boxed{\frac{1}{2} }
Question: Six people-A, B, C, D, E, and F-attend a movie premiere. The theater has 12 seats arranged in four rows, with three seats per row.The seating arrangements are determined by the following conditions: There is an empty seat directly in front of A. B is seated in the row immediately in front of A. D and A are in the same row. C and E occupy adjacent seats (left-right neighbors). A is seated in front of C. F is seated far away from A. Both B and F state that they can see the exact center of the screen. F tells B, "But I am seated farther back than you." D remarks, "Luckily, no one is blocking my view of the screen."
Rows are numbered 1 to 4, starting from the one closest to the screen. The center seat of each row (Column 2) provides a direct view of the exact center of the screen. Z_{11} is the leftmost seat in the first row, and Z_{43} is the third seat in the fourth row. Among these six people, what fraction have positions that are uniquely determined?
Answer: \boxed{\frac{1}{2} }
Question: A study group of six students-A, B, C, D, E and M (me, the narrator)-visits the library. The reading room contains 12 seats arranged in four rows, with three seats per row.The seating positions are determined by the following conditions: The seat directly in front of M is empty. A is seated in the row immediately in front of M. C is seated in the same row as M. B and D occupy adjacent seats (left-right neighbors). M and B are in the same column, with exactly one empty seat between them. E is seated far away from M. Both A and E state that their seats offer a direct view of the exact center of the window. E tells A, "But I am farther from the window than you are." C remarks, "Luckily, no one is blocking my view." 
Rows are numbered 1 to 4, with Row 1 being the closest to the window. The center seat of each row (Column 2) provides a direct line of sight to the exact center of the window. Z_{11} is the leftmost seat in the first row, and Z_{43} is the third seat in the fourth row. Among these six people, what fraction have positions that are uniquely determined?
Answer: \boxed{\frac{1}{2} }
Question: A high-speed train with a maximum capacity of 120 passengers departs from South Station carrying 60 business travelers, 5 fare-exempt children, and 2 on-board technicians; immediately prior to departure, a platform guide boards briefly to verify information and then disembarks. During the journey, 15 passengers disembark and 20 passengers board at West Garden Station; during a temporary stop due to a signaling fault, the 2 on-board technicians disembark to address the issue, and 3 passengers leave to transfer to alternative transportation; at Cloud Mountain Station, 25 passengers disembark and 12 passengers board; at an unscheduled stop, 7 passengers disembark as scheduled, and another 2 passengers are removed for having boarded the wrong train; at the penultimate station, 11 passengers disembark, and 1 railway police officer boards for duty. How many people are on board the train when it reaches its final destination?
Answer: \boxed{38}
Question: A double-decker bus with a maximum capacity of 60 people departs from the city center carrying 25 passengers, 3 fare-exempt students, and 2 photographers filming the journey; a traffic assistant boards to conduct an inspection and then leaves. During the journey, 8 people disembark and 12 people board at University Town Station; to facilitate movie filming, the 2 photographers temporarily disembark to adjust camera positions, and 1 passenger leaves via ambulance due to illness; at Tech Park Station, 11 people disembark and 6 people board; at a temporary bus stop, 4 people disembark as scheduled, and another 2 passengers are asked to leave by the driver due to a dispute; at the penultimate station, 7 people disembark, and 1 security officer boards early. How many people are on the bus when it reaches its final destination?
Answer: \boxed{17}
Question: A subway train with a maximum capacity of 500 people departs from the terminus carrying 150 passengers, 10 fare-exempt individuals, and 4 maintenance technicians; a station attendant boards briefly to verify information and disembarks before the train leaves. During the transit, 40 individuals disembark and 60 new passengers board at People's Square Station; following a signaling fault in the tunnel, the 4 technicians move to the area outside the driver's cab to address the issue, and 5 passengers exit via the emergency evacuation doors due to panic; at Century Avenue Station, 80 individuals disembark and 35 new passengers board; at a temporary station closure, 15 passengers choose to exit the station to transfer, and another 3 fare-evaders are identified and escorted off the train; at the penultimate station, 25 individuals disembark, and 1 inspector boards the train. How many people are on board when the train reaches the final destination?
Answer: \boxed{93}
Question: A standard 3x3 Rubik's Cube is initially in its solved state with the following color configuration: Front (F) is Red, Back (B) is Orange, Up (U) is Green, Down (D) is Blue, Left (L) is White, and Right (R) is Yellow. The cube undergoes two full cycles of the following sequence of moves: U, L, D, R, F, B, where each move represents a 90-degree clockwise rotation of the respective face performed in that specific order. After completing these two rounds, what is the color distribution of the Front face? Provide the resulting colors of the 3x3 grid row by row, from top-to-bottom and left-to-right.
Answer: Row 1: Yellow, Red, Blue
Question: A standard 3×3 Rubik's Cube starts with the following color configuration: Front (F) is Green, Back (B) is Green, Up (U) is Red, Down (D) is Orange, Left (L) is Blue, and Right (R) is Yellow. A single operation cycle consists of performing a 90-degree clockwise rotation on each face in the specific sequence of R, U, L, D, B, and F. After completing 4 full rounds of this sequence, what is the color distribution of the Front face? Provide the colors of the nine stickers in a 3×3 grid format, listed row by row from top-to-bottom and left-to-right.
Answer: First row: Red Green Yellow
Question: A standard 3x3 Rubik's Cube is initially in its solved state with the following color configuration: Front (F) is Yellow, Back (B) is White, Up (U) is Blue, Down (D) is Green, Left (L) is Red, and Right (R) is Orange. The cube undergoes three full rounds of a movement sequence defined as a 90-degree clockwise rotation of each face performed in the specific order of L, F, U, B, R, and D. After completing these three cycles, what is the resulting color distribution of the Front face? Provide the colors of the nine stickers in a 3x3 grid format, listed row by row from top-to-bottom and left-to-right.
Answer: Row 1: Orange, Blue, Yellow
Question: A farming cooperative provides subsidies or issues warnings on the first of each month based on the month-on-month (MoM) change in grain production from the preceding month. A subsidy is granted if production increased, while a warning is issued if production decreased. Which of the following options is/are incorrect?
A. If Farmer A's total output in the second quarter is twice that of the first quarter, the number of subsidies he receives in the second quarter must be greater than the number of warnings.
B. If Farmer B's monthly grain output shows consistent month-on-month growth throughout a specific year, he will not receive any warnings during that year.
C. If Farmer C's land lies fallow for two months every year (resulting in zero output during those months), he will receive at least one warning each year.
D. If Farmer D receives only one subsidy throughout the year, his total annual grain output for this year must be lower than that of the previous year.
Answer: ABCD
Question: An app store adjusts an app's recommendation tier on the first of each month based on the month-on-month (MoM) change in its download volume from the preceding month. An increase in downloads results in a tier upgrade, while a decrease results in a tier downgrade. Which of the following statements is/are incorrect?
A. If an app's total downloads in the second half of the year are only one-third of those in the first half, the number of recommendation tier downgrades it receives during the second half must be greater than the number of upgrades.
B. If an app achieves consistent month-on-month growth in downloads every month throughout the year, it is impossible for its recommendation tier to be downgraded during that year.
C. If an app was delisted for two months due to technical issues (resulting in zero downloads), its recommendation tier must have been downgraded at least once during that year.
D. If an app's recommendation tier was upgraded only once throughout the year, its total annual download volume for this year must be lower than that of the previous year.
Answer: ABCD
Question: On the first day of every month, a publishing house decides whether to reprint a book series based on the month-on-month (MoM) change in sales from the preceding month. If sales increased, a reprint is ordered; if sales decreased, the reprint is suspended. Which of the following statements is/are incorrect?
A. If a book's total sales in the second half of the year are only half of its total sales in the first half, the number of times the reprint was suspended in the second half must be greater than the number of times it was ordered during the same period.
B. If a book's monthly sales achieved consistent month-on-month growth throughout the year, the publisher must not have suspended any reprint during that year.
C. If a book was withdrawn from sale for three months due to content revisions (resulting in zero sales), the publisher must have suspended the reprint at least once during that year.
D. If a reprint was ordered only once during the entire year (specifically in February), the book's total annual sales for this year must be lower than those of the previous year.
Answer: ABCD
Question: On a military exercise map, a square plain is evenly divided into a 9×9 grid. From north to south, the rows are labeled 1 to 9; from west to east, the columns are labeled A to I. Under this system, the grid square in the southwest corner is labeled A9. 
A pilot receives a mission: starting from the intersection point of combat zones I9 and H8, he flies in a straight line along the shortest path to the northeast corner of combat zone F1. Upon reaching the boundary of a combat zone in Row 4, he changes course and flies in a straight line to the intersection point where combat zones B9 and C9 meet the map's edge. After conducting reconnaissance at that point, he flies along a curved path within the exercise area back to the starting point, such that the path from the reconnaissance point to the starting point forms exactly a semicircular arc. During this flight, how many distinct internal combat zones does the pilot's path pass through?
Answer: \boxed{21}
Question: On a military exercise map, a square plain is evenly divided into a 9x9 grid. From north to south, the rows are labeled 1 to 9, and from west to east, the columns are labeled A to I, such that the southwesternmost sector is designated as A9. 
A pilot is assigned a mission: starting from the intersection point of sectors I9 and H8, he flies in a straight line along the shortest path toward sector B2. Upon reaching the boundary of Row 4, he changes course and flies in a straight line toward the point where the boundary between sectors B9 and C9 meets the edge of the map. After conducting reconnaissance at that junction, he flies back to the initial departure point along a path within the exercise area that forms a perfect semicircular arc. During this entire flight, how many distinct internal sectors does the pilot's path pass through?
Answer: \boxed{15}
Question: On a military exercise map, a square plain is evenly divided into a 9x9 grid. Rows are labeled 1 to 9 from north to south, and columns are labeled A to I from west to east, such that the southwesternmost sector is designated as A9. 
A pilot is assigned a mission: starting from the intersection point of sectors I9 and H8, he flies in a straight line along the shortest path toward sector C3. Upon reaching the boundary of Row 4, he changes course and flies in a straight line toward the intersection point where the boundary between sectors B9 and C9 meets the edge of the map. After conducting reconnaissance at that junction, he returns to the initial departure point along a path within the exercise area that forms a perfect semicircular arc. During this entire flight, how many distinct internal sectors does the pilot's path pass through?
Answer: \boxed{15}
Question: Detective Harris is interrogating a high-profile jewel thief in an attempt to crack the passcode to a safe where stolen goods are hidden. The passcode consists of four distinct digits (0–9). Detective Harris performs four trial entries, and after each attempt, the system provides feedback in the XAYB format (where A represents the number of digits correct in both value and position, and B represents the number of digits correct in value but in the wrong position).
Based on the feedback from the four trials below, deduce the correct four-digit passcode:
First Attempt: 1234 - Feedback: 0A2B
Second Attempt: 5678 - Feedback: 0A2B
Third Attempt: 3456 - Feedback: 1A0B
Fourth Attempt: 7812 - Feedback: 1A2B
What is the correct passcode?
Answer: \boxed{7481}
Question: The following is a logic puzzle based on the game 1A2B. The codemaker sets a secret four-digit code, and the codebreaker attempts to guess it. For each guess, the setter provides a hint in the format of XAYB, where X (A) represents the number of digits that are correct in both value and position, and Y (B) represents the number of digits that are correct in value but in the wrong position. The game continues until the code is guessed correctly.
Welcome to the Guess the Number game! Please guess a 4-digit number where each digit is unique.
Guess: 1234 - Result: 0A1B
Guess: 5678 - Result: 0A1B
Guess: 1357 - Result: 1A0B
Guess: 4098 - Result: 3A0B
Based on the logs above, what is the correct secret code?
Answer: \boxed{4097}
Question: The following is a game of 1A2B. The codemaker sets a secret four-digit code, and the codebreaker tries to guess it. For each guess, the codemaker provides a hint in the format of XAYB, where X (A) represents the number of digits that are correct in both value and position, and Y (B) represents the number of digits that are correct in value but in the wrong position. The game continues until the code is guessed correctly.
Welcome to the Guess the Number game! Please guess a 4-digit number where each digit is unique.
Please enter your guess: 1234 - Result: 1A1B 
Please enter your guess: 5678 - Result: 0A1B 
Please enter your guess: 9012 - Result: 0A3B 
Please enter your guess: 1927 - Result: 2A2B
Based on the attempts above, what is the correct code?
Answer: \boxed{1729}
Question: A rigid figure-eight wire loop, formed by two identical standard circles welded together at their point of tangency, is interlocked with a standard torus. One of the circular sub-loops passes through the central hole of the torus so that they are linked. Given that the wire loop and the surface of the torus can touch, what is the maximum possible number of contact points between them?
Answer: \boxed{4}
Question: A movable standard circular wire ring is interlocked with an elliptical torus (formed by revolving a standard ellipse). The two are linked through each other's central holes such that they cannot be separated. Given that the circular ring and the surface of the elliptical torus have points of contact, what is the maximum possible number of contact points between them?
Answer: \boxed{4}
Question: A standard circular wire ring is interlocked with a thick metal washer (a flat circular disc with a central hole) such that they cannot be separated. Given that the ring is in partial contact with the surface of the washer (there are contact points, but the ring is not in full contact), what is the maximum possible number of contact points between them?
Answer: \boxed{4}
Question: Consider a 9-digit number formed by seven distinct digits, represented by the pattern "abcdaefgd", which satisfies the following conditions:
1. The two-digit number ab is a perfect square such that ab=c².
2. The two-digit number fg satisfies the equation fg=c*d-e.
3. The digits e, d, and a (in that order) form an arithmetic progression.
4. The digit d is the smallest composite number.
What is this 9-digit number?
Answer: \boxed{819480364}
Question: Consider a 9-digit number composed of seven distinct digits, following the pattern abcdbcefg. The following conditions hold:
1. The two-digit number de is a perfect square such that de=f².
2. The digit g satisfies the equation g=b+c.
3. The digits a, b, and c (in that order) form an arithmetic progression.
4. The digit a is the unique even prime number.
What is this 9-digit number?
Answer: \boxed{234834197}
Question: Consider a 9-digit number composed of seven distinct digits, represented by the pattern abcdaefgd. The following conditions hold:
1. The two-digit number ab is a perfect square such that ab=c².
2. The two-digit number fg satisfies the equation fg=b*c-d+a.
3. The digits d, e, and a (in that order) form an arithmetic progression.
4. The digit d is 5.
What is this 9-digit number?
Answer: \boxed{164513205}
Question: A historical novelist, who is a fervent fan of Roman culture, has encrypted his most important novel outline file. The system hint indicates that the password is derived from the string of symbols "-|_" by rotating them to form letters that represent Roman numerals. What is the password?
Answer: \boxed{111}
Question: An archaeologist with an obsession for Ancient Roman history has discovered an encrypted digital map archive marking the location of a lost Roman Legion's treasure. Decrypting the archive requires a dynamic passcode. Reportedly, this passcode is derived by rotating the sequence of symbols "/\\+\\/" to form letters, which are then converted into their corresponding numerical value. Do you know what the passcode is?
Answer: \boxed{5105}
Question: A winemaker renowned for his vintage reserve wine "Bacchus" keeps his secret recipe locked inside an electronic safe. The passcode to the safe is obtained by deciphering the string of symbols "><-|". The method involves rotating these symbols to form letters that represent Roman numerals. What is the password?
Answer: \boxed{5511}
Question: Six writers-L, M, N, O, P, and Q-are arranged in a circle facing the center. The positioning rules are as follows: M is to the left of L; N is positioned directly opposite M; L is to the right of M; O is not adjacent to P; and Q cannot be adjacent to both L and N simultaneously. If Q is assigned to Position 1, and the positions are numbered 1 through 6 in a clockwise direction, what is the average position number of L across all possible valid distributions?
Answer: \boxed{4.5}