Editing Sprague–Grundy theorem (section)

==Development==
If <math>G</math> is a position of an impartial game, the unique integer <math>m</math> such that <math>G \approx *m</math> is called its Grundy value, or Grundy number, and the function that assigns this value to each such position is called the Sprague–Grundy function.  R. L. Sprague and P. M. Grundy independently gave an explicit definition of this function, not based on any concept of equivalence to nim positions, and showed that it had the following properties:
*The Grundy value of a single nim pile of size <math>m</math> (i.e. of the position <math>*m</math>) is <math>m</math>;
* A position is a loss for the next player to move (i.e. a <math>\mathcal{P}</math>-position) if and only if its Grundy value is zero; and
*The Grundy value of the sum of a finite set of positions is just the [[nim-sum]] of the Grundy values of its summands.
It follows straightforwardly from these results that if a position <math>G</math> has a Grundy value of <math>m</math>, then <math>G + H</math> has the same Grundy value as  <math>*m + H</math>, and therefore belongs to the same outcome class, for any position <math>H</math>.   Thus, although Sprague and Grundy never explicitly stated the theorem described in this article, it follows directly from their results and is credited to them.<ref>{{citation
| last= Smith
| first = Cedric A.B.
| title = Patrick Michael Grundy, 1917–1959
| journal = Journal of the Royal Statistical Society, Series A
| year = 1960
| volume = 123
| issue = 2
| pages = 221–22
}}</ref><ref>{{Cite journal|author1=Schleicher, Dierk |author2=Stoll, Michael | title = An introduction to Conway's games and numbers
 |journal=Moscow Mathematical Journal
 |volume=6 |issue=2 |pages=359–388 | year = 2006
 | arxiv = math.CO/0410026|doi=10.17323/1609-4514-2006-6-2-359-388 |s2cid=7175146 }}</ref>
These results have subsequently been developed into the field of [[combinatorial game theory]], notably by [[Richard K. Guy|Richard Guy]], [[E. R. Berlekamp|Elwyn Berlekamp]], [[John Horton Conway]] and others, where they are now encapsulated in the Sprague–Grundy theorem and its proof in the form described here. The field is presented in the books ''[[Winning Ways for your Mathematical Plays]]'' and ''[[On Numbers and Games]]''.