Upon opening the puzzle, we see different rooms, stylized like For the Win. The only room unlocked at the beginning is room 1. Clicking room 1 gives a list of five randomly generated math problems; however, one may notice that submitting the correct answer does not work. For each question, we see the responses of Bob and Charlie are 2 less and 3 less than the actual correct answer, respectively. As we play as Alex, this suggests that we should submit an answer that is 1 less than the actual answer. Solving at least 3 out of the 5 questions in this manner should lets us pass room 1, and rooms 2 and 3 become unlocked.
The other rooms are similar. Each room consists of randomly generated questions, and we compete against two other people. Each person interprets the question in a different way and gives a different answer. Using the answers and the modifications of the other people in the room, it suggests how we should change our answer to get the question correct in the room. We list all of the rooms and their corresponding modifications:
| Room Number | 2nd Player Name | 3rd Player Name | 2nd Player Modification | 3rd Player Modification | Our Answer |
|---|---|---|---|---|---|
| 1 | Bob | Charlie | Submit 2 less than the correct answer | Submit 3 less than the correct answer | Submit 1 less than the correct answer (B = -2, C = -3, so A = -1) |
| 2 | Bob | Charlie | Add 1 to the second number in the problem before computing answer | Add 1 to the third number in the problem before computing answer | Add 1 to the first number in the problem before computing answer |
| 3 | Bob | Charlie | Reverse digits of the second number in the problem before computing answer | Reverse digits of the third number in the problem before computing answer | Reverse digits of the the first number in the problem before computing answer |
| 4 | Lex | Val | Permute numbers in the problem so they are in lexicographical order, then solve | Permute numbers in the problem so they are in numerical order (ordered by value), then solve | Permute numbers in the problem so they are in alphabetical order (by English name), then solve |
| 5 | Austin | Amy | Treat every number as base 6, including final answer | Treat every number as base 3, including final answer | Treat every number as base 4, including final answer (name length = base) |
| 6 | Holly | Andrew | Compute the harmonic mean | Compute the arithmetic mean | Compute the geometric mean |
| 7 | Austin | Amy | Assume the polygon is a hexagon | Assume the polygon is a triangle | Assume the polygon is a square (name length = number of sides of polygon) |
| 8 | Aann | Anna | Transform all one digit numbers A to AA and all two digit numbers AB to AABB, then solve | Transform all one digit numbers A to AA and all two digit numbers AB to ABBA, then solve | Transform all one digit numbers A to AA and all two digit numbers AB to ABAB, then solve |
See the appendix1 for solutions to all of the generic question types.
After clearing 4 out of the 8 rooms, we unlock the final room. Here we have 8 questions. We apply one of the transformations to each of the questions in the final room to solve them. We list the questions in the final room and solutions:
| # | Question | Orig. Room | Answer | Solution |
|---|---|---|---|---|
| 1 | If and then can be expressed in simplest form as Compute | 3 | 180 | reversed is , so we want so |
| 2 | Let and and let be the largest integer less than or equal to Let be the result of removing the first digit from What is ? | 8 | 13 | becomes becomes so is then the greatest integer less than or equal to or Then is |
| 3 | How many positive factors does the number have? | 4 | 9 | We resort the numbers in the order (four), (one hundred five) and (twenty-one). Then we wish to find the factors of , which has factors. |
| 4 | How many prime numbers are there less than ? | 1 | 3 | There are prime numbers less than , so we submit one less than that, or . |
| 5 | An outfit consists of a shirt, a pair of pants, and a pair of shoes. If you have shirt, pairs of pants, and pairs of shoes, how many possible outfits can you create? | 2 | 18 | We add one to the first number appearing in the problem, so we replace with . Then there are outfits. |
| 6 | Compute the mean of the set | 6 | 10 | The geometric mean of the set is |
| 7 | How many nonnegative integers consisting solely of the digits , , or are less than or equal to ? | 5 | 12 | The valid numbers less than or equal to (in base 4) are , , , , , and . There are (in base 4) such numbers. |
| 8 | In a regular polygon, the largest distance between any two points on the polygon is What is the square of the side length of the polygon? | 7 | 10 | In a square, given side length , the largest distance between any two points is Thus, the side length of the square is The square of the side length is then |
Clearing the final room doesn’t unlock anything. However, in the final room, we do not compete against anyone, unlike the previous room. This suggests figuring out what the people we competed against in the previous room would have answered. Computing all of the answers reveals that they all lie between 1 and 26, inclusive, so we replace 1-26 with A-Z.
| # | Question | Orig. Room | Name | Ans | Char | Solution |
|---|---|---|---|---|---|---|
| 1 | If and then can be expressed in simplest form as Compute | 3 | Bob | 6 | F | so |
| Charlie | 18 | R | so | |||
| 2 | Let and and let be the largest integer less than or equal to Let be the result of removing the first digit from What is ? | 8 | Aann | 5 | E | so and |
| Anna | 14 | N | so and | |||
| 3 | How many positive factors does the number have? | 4 | Lex | 3 | C | so there are factors. |
| Val | 8 | H | so there are factors. | |||
| 4 | How many prime numbers are there less than ? | 1 | Bob | 2 | B | |
| Charlie | 1 | A | ||||
| 5 | An outfit consists of a shirt, a pair of pants, and a pair of shoes. If you have shirt, pairs of pants, and pairs of shoes, how many possible outfits can you create? | 2 | Bob | 12 | L | |
| Charlie | 12 | L | ||||
| 6 | Compute the mean of the set | 6 | Holly | 5 | E | The harmonic mean is |
| Andrew | 20 | T | The arithmetic mean is | |||
| 7 | How many nonnegative integers consisting solely of the digits , , or are less than or equal to ? | 5 | Andrew | 4 | D | The valid numbers less than or equal to (in base 6) are and There are (in base 6) such numbers. |
| Amy | 21 | U | The valid numbers less than or equal to (in base 3) are and There are (in base 3) such numbers. | |||
| 8 | In a regular polygon, the largest distance between any two points on the polygon is What is the square of the side length of the polygon? | 7 | Andrew | 5 | E | In a hexagon, given side length , the largest distance between any two points is Thus, the side length of the square is The square of the side length is then |
| Amy | 20 | T | In a triangle, given side length , the largest distance between any two points is . Thus, the side length of the square is The square of the side length is then |
Reading the letters row by row gives the clue phrase FRENCH BALLET DUET, which solves to PAS DE DEUX.
Author's Notes
I never played a lot of For the Win! in middle and high school, but a lot of my friends did. I wanted to write a puzzle about a math contest, and For the Win was a pretty natural choice for something that would be lots of fun.
The first modification I thought of was the base one; I was inspired by the Fun With Numbers puzzle from 2010 Mystery Hunt and an old mock 2010 AIME I did back when I did contests. I also drew inspiration from NIMO April Fun rounds (which I can’t seem to find anymore, sad).
Thanks to Justin for suggesting room ideas, Herman, Joanna, and Brian for implementing the web interface, and Eggy and Ciel for being my editors and suggesting the final extraction.
Appendix: Generic problem solutions
| Question | Answer | Notes |
|---|---|---|
| An outfit consists of a shirt, a pair of pants, and a pair of shoes. If you have shirts, pairs of pants, and pairs of shoes, how many possible outfits can you create? | ||
| Compute | ||
| Alice, Bob, Carl, and Diane all lie on a line. Alice and Bob are meters apart, Bob and Carl are meters apart, and Carl and Diane are meters apart. What is the positive difference between the minimum and maximum distances between Alice and Diane? | ||
| Compute | ||
| The roots of a monic cubic polynomial are and What is the constant coefficient of the polynomial? | See Vieta’s formulas | |
| A cone has radius and height If the cone is scaled up so that the radius and height are each multiplied by then the volume of the new cone is What is ? | ||
| If , , and , what is ? | ||
| Let , and let . If can be expressed in simplest terms as , compute . | ||
| Compute . | ||
| Compute . | ||
| Compute . | ||
| Compute . | ||
| Compute . | ||
| Compute . | ||
| Compute the number of positive factors of . | See Divisor function | |
| Compute the number of positive factors of . | See Divisor function | |
| Compute the number of positive factors of . | See Divisor function | |
| Compute the number of positive factors of . | See Divisor function | |
| Compute the number of positive factors of . | See Divisor function | |
| There are terms in an arithmetic sequence, starting with . If is the common difference, what is the last term in the sequence? | ||
| There are terms in a geometric sequence, starting with . If is the common ratio, what is the last term in the sequence? | ||
| What percentage of is of ? | ||
| of what number is of ? | ||
| Trapezoid has , , and , with right angles at and . Compute the value of . | ||
| In a box, there are forks, spoons, and knives. Alice grabs utensils uniformly at random, a single utensil at a time, until she has at least a fork, a spoon, and a knife. What is the positive difference between the maximum and minimum number of utensils that Alice grabs? | ||
| In triangle , , , and . If is the area of triangle , compute . | See Heron’s formula | |
| A regular polygon has side length . What is the area of the polygon? | Square: Hexagon: Triangle: | |
| A regular polygon has side length . What is the perimeter of the polygon? | Square: Hexagon: Triangle: | |
| A regular polygon has side length . What is the largest distance between any two points on the polygon? | Square: Hexagon: Triangle: | |
| A regular polygon has side length . If the area of the smallest circle that surrounds the polygon is , what is ? | Square: Hexagon: Triangle: | See Circumscribed circle |
| A regular polygon has side length . If the area of the largest circle that contains the polygon is , what is ? | Square: Hexagon: Triangle: | See Incircle |
| Find the mean of . | variable | See Pythagorean means |