Editing Sprague–Grundy theorem (section)

===Combining Games===
Two games can be combined by '''''adding''''' their positions together.
For example, consider another game of nim with heaps <math>A'</math>, <math>B'</math>, and <math>C'</math>.

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

We can combine it with our [[#firstExample|first example]] to get a combined game with six heaps: <math>A</math>, <math>B</math>, <math>C</math>, <math>A'</math>, <math>B'</math>, and <math>C'</math>:

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

To differentiate between the two games, for the [[#firstExample|first example game]], we'll label its starting position <math>\color{blue}S</math>, and color it blue:
<math display="block">\color{blue}S = \Big\{
    \big\{*1, \{*1\}, *2\big\},
    \big\{*2, \{*1, \{*1\},*2\} \big\},
    \big\{\{*1\}, \{\{*1\}\}, \{*1, \{*1\}, *2\}\big\}
\Big\}</math>

For the [[#secondExample|second example game]], we'll label the starting position <math>\color{red}S'</math> and color it red:
<math display="block">
\color{red}S' = \Big\{\{*1\}\Big\}.
</math>

To compute the starting position of the [[#thirdExample|combined game]], remember that a player can either make a move in the first game, leaving the second game untouched, or make a move in the second game, leaving the first game untouched.  So the combined game's starting position is:
<math display="block">
\color{blue}S \color{black} + \color{red}S' \color{black}= 
\Big\{
    \color{blue}S\color{black} + \color{red} \{*1\} \color{black}
\Big\} \cup \Big\{
     \color{red}S'\color{black} + \color{blue}\{*1, \{*1\}, *2\} \color{black}, 
     \color{red}S'\color{black} + \color{blue} \{*2, \{*1, \{*1\},*2\} \} \color{black},
     \color{red}S'\color{black} + \color{blue} \{\{*1\}, \{\{*1\}\}, \{*1, \{*1\}, *2\}\} \color{black}
\Big\}
</math>

The explicit formula for adding positions is: <math>S+S'=\{S+s'\mid s'\in S'\}\cup\{s+S'\mid s\in S\}</math>, which means that addition is both commutative and associative.