## Nim

### Variations

There are many variations of Nim, most of which can be analysed mathematically in a similar way to Nim.

**Misère Nim**

Misère Nim is identical to Nim, except that the player to take the last dot loses.

**The Subtraction Game**

The Subtraction Game starts with one or more heaps, like Nim, but there is a limit on the number of objects that each player may remove. For example, the game might start with a single heap of 10 objects, and each player is allowed to remove up to 3 objects.

**Grundy's game**

Grundy's game starts with a single heap, such as of 10 objects, and the players take turns in dividing the heap into two heaps of different sizes. Thus, a heap of six objects may be divided into heaps of 5+1 or 4+2, but not 3+3.