Editing Sprague–Grundy theorem (section)

=== {{anchor|firstExample}}Example Nim Game ===
{{Pre|
 Sizes of heaps  Moves
 A B C
 &nbsp;
 1 2 2           Alice takes 1 from A
 0 2 2           Bob   takes 1 from B 
 0 1 2           Alice takes 1 from C 
 0 1 1           Bob   takes 1 from B 
 0 0 1           Alice takes 1 from C
 0 0 0           Bob   has no moves, so Alice wins
}}

* At step 6 of the game (when all of the heaps are empty) the position is <math>\{\}</math>, because Bob has no valid moves to make.  We name this position <math>*0</math>. 
* At step 5, Alice had exactly one option: to remove one object from heap C, leaving Bob with no moves.  Since her ''move'' leaves Bob in position <math>*0</math>, her ''position'' is written <math>\{ *0 \}</math>.  We name this position <math>*1</math>.
* At step 4, Bob had two options: remove one from B or remove one from C.  Note, however, that it didn't really matter which heap Bob removed the object from: Either way, Alice would be left with exactly one object in exactly one pile. So, using our recursive definition, Bob really only has one move: <math>*1</math>.   Thus, Bob's position is <math>\{*1\}</math>.
* At step 3, Alice had 3 options: remove two from C, remove one from C, or remove one from B.  Removing two from C leaves Bob in position <math>*1</math>.  Removing one from C leaves Bob with two piles, each of size one, i.e., position <math>\{*1\}</math>, as described in step 4. However, removing 1 from B would leave Bob with two objects in a single pile. ''His'' moves would then be <math>*0</math> and  <math>*1</math>, so ''her'' move would result in the position <math>\{*0, *1\}</math>.  We call this position <math>*2</math>.  Alice's position is then the set of all her moves: <math>\big\{*1, \{*1\}, *2\big\}</math>.
* Following the same recursive logic, at step 2, Bob's position is <math display="block">\big\{ \{*1, \{*1\}, *2\}, *2\big\}.</math>
* Finally, at step 1, Alice's position is <math display="block">\Big\{
    \big\{*1, \{*1\}, *2\big\},
    \big\{*2, \{*1, \{*1\},*2\} \big\},
    \big\{\{*1\}, \{\{*1\}\}, \{*1, \{*1\}, *2\}\big\}
\Big\}.</math>