Heads point outward; numbers are inside heads. For a large group, zoom and pan within the arena. Instructions scroll separately.
Understand the answer, step by step
1. Try five players
| Token holder | Removed | Passes to |
|---|
| 1 | 2 | 3 |
| 3 | 4 | 5 |
| 5 | 1 | 3 |
| 3 | 5 | 3 wins |
Player 3 survives. Notice the wrap from 5 to 1 and that removed players are skipped.
2. Find the special group sizes
With 2, 4, 8 or 16 players, the starting token holder wins. These are powers of two. For example, with 8, the first round removes 2, 4, 6 and 8. Four players remain, starting again at 1. The same pattern repeats.
3. Reduce a larger number to a power of two
For 100, the largest power of two not exceeding 100 is 64. There are 100 − 64 = 36 extra players. After 36 turns, players 2, 4, …, 72 have been removed. There are now exactly 64 players left, and player 73 holds the token. The starting holder of a power-of-two group wins, so 73 survives.
4. Write the shortcut
Let P be the largest power of 2 no larger than n. Let L = n − P. Then the winning original position is 2L + 1. The token has advanced two original positions for each of those L removals.
| n | P | L | 2L + 1 |
|---|
| 5 | 4 | 1 | 3 |
| 10 | 8 | 2 | 5 |
| 100 | 64 | 36 | 73 |
| 200 | 128 | 72 | 145 |
This shortcut applies to this exact remove-every-second rule, starting with player 1 and removing player 2 first. Change the passing rule, starting player or direction, and the result can change.