Editing Nim (section)

=== The 21 game ===
{{See also|21 (drinking game)}}
The game "21" is played as a misère game with any number of players who take turns saying a number. The first player says "1" and each player in turn increases the number by 1, 2, or 3, but may not exceed 21; the player forced to say "21" loses. This can be modeled as a subtraction game with a heap of {{nowrap|21 − ''n''}} objects. The winning strategy for the two-player version of this game is to always say a multiple of 4; it is then guaranteed that the other player will ultimately have to say 21; so in the standard version, wherein the first player opens with "1", they start with a losing move.

The 21 game can also be played with different numbers, e.g., "Add at most 5; lose on 34".

A sample game of 21 in which the second player follows the winning strategy:

{| class="wikitable"
! Player !! Number
|-
| 1 || 1
|-
| 2 || 4
|-
| 1 || 5, 6 or 7
|-
| 2 || 8
|-
| 1 || 9, 10 or 11
|-
| 2 || 12
|-
| 1 || 13, 14 or 15
|-
| 2 || 16
|-
| 1 || 17, 18 or 19
|-
| 2 || 20
|-
| 1 || 21
|}