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Games & Processes

Turn-based games, iterated processes, tournaments, scheduling

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Games & ProcessesAMC 10 Step by Step

Name: Date: October 28, 2026

14 problems, easiest first

What you need to know

Round-robin. If $n$ teams each play every other team once, there are $\binom{n}{2}$ games. With no ties, each game has one winner, so the wins also add up to $\binom{n}{2}$ . Example: $8$ teams play $\binom82 = 28$ games and win $28$ in all.
Single elimination. $n$ players need exactly $n-1$ matches: each match knocks out one player, and everyone but the champion is knocked out. Example: $64$ players need $63$ matches.
Invariants. An invariant is something no step changes, like even/odd, or a total's remainder when divided by some $m$ . A final state with a different value can never be reached. Example: $1, 2, \dots, 10$ are on a board. A move erases two numbers and writes their difference, which lowers the sum by twice the smaller one. So the sum, $55$ at the start, stays odd, and the last number left is never $0$ .
Winning and losing positions. A position is losing if the player to move loses with best play on both sides. That happens when every move leads to a winning position. A position is winning when some move leads to a losing one. In "take $1$ to $k$ stones; whoever takes the last stone wins," the losing positions are the multiples of $k+1$ . Example: taking $1$ to $3$ stones, $20$ is losing. Whatever you take, your opponent tops the round up to $4$ stones, and in the end takes the last one.

How AMC 10 tests it

Problems 5–12: games in a tournament, handshakes at a party, days until a schedule repeats.
Problems 12–20: a number is changed again and again by a rule; find its value after many steps.
Problems 15–22: two-player games of taking objects away, asking which starting sizes are first-player wins.
Problems 18–25: a repeated process with a counting answer.

Standard approaches

Work out the first several steps by hand; a cycle or pattern usually shows up fast.
Look for an invariant the process keeps (even/odd, a sum's remainder).
For two-player games, work backward from the end, marking positions W (winning) or L (losing).
For tournaments, use total wins $= \binom{n}{2}$ to set up an equation.
For "how many starting values," find the pattern, then count with remainders.

Worked example

Alice and Bob alternately remove $1$ , $3$ , or $4$ coins from a pile of $n$ coins, Alice first; whoever takes the last coin wins. For how many integers $n$ with $1 \le n \le 100$ does Bob have a winning strategy?

(A) $14$ (B) $20$ (C) $28$ (D) $29$ (E) $34$

Solution.

Step 1: the rule. Mark each pile size L (losing) or W (winning) for the player about to move. A pile of $0$ is L (the other player just took the last coin). A pile is W exactly when taking $1$ , $3$ , or $4$ coins can leave an L pile.

Step 2: the table. For example, $2$ is L (its only move leaves $1$ , a W), and $7$ is L (its moves leave $6$ , $4$ , $3$ , all W).

$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$	$16$
W/L	L	W	L	W	W	W	W	L	W	L	W	W	W	W	L	W	L

Step 3: the period. Each entry depends only on the entries $1$ , $3$ , and $4$ places back. Entries $7$ to $10$ (L, W, L, W) match entries $0$ to $3$ , so from there on the table repeats every $7$ . The L piles are the numbers with remainder $0$ or $2$ when divided by $7$ (written $n \equiv 0$ or $2 \pmod 7$ ).

Step 4: who wins. Alice moves first, so Bob wins exactly when the starting pile is L.

Step 5: count. Multiples of $7$ : $7, 14, \dots, 98$ , which is $14$ numbers. Remainder $2$ : $2, 9, \dots, 100$ , which is $\frac{100-2}{7}+1 = 15$ numbers. Total: $14+15 = 29$ .

The answer is $\boxed{\textbf{(D)}\ 29}$ .

Pitfalls

Counting round-robin games as $n(n-1)$ instead of $\binom{n}{2}$ . That counts each game twice: $8$ teams play $28$ games, not $56$ .
Stopping a simulation before the cycle has repeated in full, then following a false pattern.
Marking a position W because one move leads to a W position. A winning move must lead to an L position.
Off-by-one mistakes between "after $k$ steps" and "at step $k$ ." If the start is step $1$ , then after $5$ steps you are at step $6$ .

Traps that recur

Treating the game as ordinary Nim (using wall sizes 6, 2, 1 directly) even though removing a middle brick splits a wall into two. (2021 AMC 10B #24)
Treating the 18 games as freely arrangeable (choosing which 3 go in each round) without enforcing that every player plays exactly once per round. (2013 AMC 10A #24)
Reporting the number of transitive triples, 945 (choice C), or the total number of triples, 1330 (choice E), instead of their difference. (2016 AMC 10B #22)
Trying to simulate all the exchanges by hand, or assuming the process ends with 0 tokens of each color. (2013 AMC 10B #17)

1.2024 AMC 10A #6
What is the minimum number of successive swaps of adjacent letters in the string $ABCDEF$ that are needed to change the string to $FEDCBA?$ (For example, $3$ swaps are required to change $ABC$ to $CBA;$ one such sequence of swaps is $ABC\to BAC\to BCA\to CBA.$ )
1. A) $6$
2. B) $10$
3. C) $12$
4. D) $15$
5. E) $24$
2.2017 AMC 10A #4
Mia is “helping” her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys into the box for the first time?
1. A) $13.5$
2. B) $14$
3. C) $14.5$
4. D) $15$
5. E) $15.5$
3.2004 AMC 10A #8
A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players $A$ , $B$ , and $C$ start with $15$ , $14$ , and $13$ tokens, respectively. How many rounds will there be in the game?
1. A) $36$
2. B) $37$
3. C) $38$
4. D) $39$
5. E) $40$
4.2003 AMC 10B #15
There are $100$ players in a single tennis tournament. The tournament is single elimination, meaning that a player who loses a match is eliminated. In the first round, the strongest $28$ players are given a bye, and the remaining $72$ players are paired off to play. After each round, the remaining players play in the next round. The match continues until only one player remains unbeaten. The total number of matches played is
1. A) $\text{a prime number}$
2. B) $\text{divisible by 2}$
3. C) $\text{divisible by 5}$
4. D) $\text{divisible by 7}$
5. E) $\text{divisible by 11}$
5.2024 AMC 10B #16
Jerry likes to play with numbers. One day, he wrote all the integers from $1$ to $2024$ on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them by either their sum or their product. (For example, Jerry's first step might have been to erase $1$ , $2$ , $3$ , and $5$ , and then write either $11$ , their sum, or $30$ , their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the whiteboard were odd. What is the maximum possible number of integers on the whiteboard at that time?
1. A) $1010$
2. B) $1011$
3. C) $1012$
4. D) $1013$
5. E) $1014$
6.2023 AMC 10A #16
In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?
1. A) $15$
2. B) $36$
3. C) $45$
4. D) $48$
5. E) $66$
7.2020 AMC 10B #16
Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than $4$ . They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$ . Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?
1. A) $\text{ Bela will always win.}$
2. B) $\text{ Jenn will always win.}$
3. C) $\text{ Bela will win if and only if }n \text{ is odd.}$
4. D) $\text{ Jenn will win if and only if }n \text{ is odd.}$
5. E) $\text { Jenn will win if and only if } n>8.$
8.2012 AMC 10B #20
Bernardo and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernardo. The winner is the last person who produces a number less than 1000. Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N$ ?
1. A) $7$
2. B) $8$
3. C) $9$
4. D) $10$
5. E) $11$
9.2012 AMC 10B #15
In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of the tournament?
1. A) $2$
2. B) $3$
3. C) $4$
4. D) $5$
5. E) $6$
10.2016 AMC 10B #22
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$ , $B$ beat $C$ , and $C$ beat $A?$
1. A) $385$
2. B) $665$
3. C) $945$
4. D) $1140$
5. E) $1330$
11.2013 AMC 10B #17
Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?
1. A) $62$
2. B) $82$
3. C) $83$
4. D) $102$
5. E) $103$
12.2011 AMC 10A #23
Seven students count from 1 to 1000 as follows:
- Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, ..., 997, 999, 1000.
- Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
- Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
- Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
- Finally, George says the only number that no one else says.
What number does George say?
1. A) $37$
2. B) $242$
3. C) $365$
4. D) $728$
5. E) $998$
13.2021 AMC 10B #24
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move: $(3,2),(2,1,2),(4),(4,1),(2,2),$ or $(1,1,2).$

Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?
1. A) $(6,1,1)$
2. B) $(6,2,1)$
3. C) $(6,2,2)$
4. D) $(6,3,1)$
5. E) $(6,3,2)$
14.2013 AMC 10A #24
Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?
1. A) $540$
2. B) $600$
3. C) $720$
4. D) $810$
5. E) $900$

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.