Topics / Counting & Probability

Games & Processes

Turn-based games, iterated processes, tournaments, scheduling

Games & ProcessesAMC 10 Step by Step

Name: Date: September 13, 2026

Counting & Probability · 14 problems have this as their main idea

primary-topic problems (1.1% of all)

more as a secondary topic

Where it appears

P1-10

P11-15

P16-20

P21-25

## What you need to know
- Round-robin: $n$ teams each playing every other once play $\binom{n}{2}$ games, which is also the total number of wins.
- Single elimination: $n$ players need exactly $n - 1$ matches; each match eliminates one player.
- Invariants: a parity or residue preserved by every step rules out impossible final states.
- Winning and losing positions: a position is losing if every move leads to a winning position, winning if some move leads to a losing one; in "remove $1$ to $k$ stones," multiples of $k + 1$ are losing.

## 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 repeatedly transformed by a rule; find its value after many steps.
- Problems 15–22: two-player take-away games, asking which starting sizes are first-player wins.
- Problems 18–25: an iterated process with a counting answer.

## Standard approaches
1. Simulate the first several steps; a cycle or pattern usually appears quickly.
2. Look for an invariant (parity, sum mod $m$ ) the process preserves.
3. For two-player games, work backward from the end, labeling positions W or L.
4. For tournaments, use total wins $= \binom{n}{2}$ to set up an equation.
5. For "how many starting values," find the pattern and count with divisibility.

## 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. Call $n$ losing if the player to move loses under best play. $0$ is losing; $n$ is winning exactly when some move lands on a losing position. Computing upward: $1$ W, $2$ L, $3$ – $6$ W, $7$ L, $8$ W, $9$ L, $10$ – $13$ W, $14$ L, $16$ L. Since the status of $n$ depends only on $n - 1$ , $n - 3$ , $n - 4$ , the pattern has period $7$ : losing positions are $n \equiv 0$ or $2 \pmod 7$ . Bob wins exactly when the start is losing. For $1 \le n \le 100$ , multiples of $7$ give $14$ values and $n \equiv 2 \pmod 7$ gives $15$ . Total $29$ , answer $\boxed{\textbf{(D)}\ 29}$ .

## Pitfalls
- Counting round-robin games as $n(n-1)$ instead of $\binom{n}{2}$ .
- Stopping a simulation before the cycle has clearly repeated and extrapolating a false pattern.
- Labeling a position W because one move leads to W; it must lead to an L position.
- Off-by-one errors between "after $k$ steps" and "at step $k$ ."

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)

Problems, easiest first

2024 AMC 10A · #6Games & Processes

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 $ABC\to BAC\to BCA\to CBA.$ )

2017 AMC 10A · #4Games & Processes

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 …

2004 AMC 10A · #8Games & Processes

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 $13$ …

2003 AMC 10B · #15Games & Processes

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 …

2024 AMC 10B · #16Games & Processes

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 $5$ , …

2023 AMC 10A · #16Games & Processes

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 …

2020 AMC 10B · #16Games & Processes

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 …

2012 AMC 10B · #20Games & Processes

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 …

2012 AMC 10B · #15Games & Processes

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 …

2016 AMC 10B · #22Games & Processes

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 $A?$

2013 AMC 10B · #17Games & Processes

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 …

2011 AMC 10A · #23Games & Processes

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 …

2021 AMC 10B · #24Games & Processes

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 …

2013 AMC 10A · #24Games & Processes

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 …