Editing Combinatorial game theory (section)

==Examples==
The introductory text ''[[Winning Ways]]'' introduced a large number of games, but the following were used as motivating examples for the introductory theory:
* Blue–Red [[Hackenbush]] - At the finite level, this partisan combinatorial game allows constructions of games whose values are [[Dyadic rational|dyadic rational number]]s. At the infinite level, it allows one to construct all real values, as well as many infinite ones that fall within the class of [[surreal numbers]].
* Blue–Red–Green Hackenbush - Allows for additional game values that are not numbers in the traditional sense, for example, [[star (game theory)|star]].
* [[Toads and Frogs]] - Allows various game values. Unlike most other games, a position is easily represented by a short string of characters.
* [[Domineering]] - Various interesting games, such as [[hot game]]s, appear as positions in Domineering, because there is sometimes an incentive to move, and sometimes not.  This allows discussion of a game's [[temperature (game theory)|temperature]].
* [[Nim]] - An [[impartial game]]. This allows for the construction of the [[nimber]]s.  (It can also be seen as a green-only special case of Blue-Red-Green Hackenbush.)

The classic game [[Go (board game)|Go]] was influential on the early combinatorial game theory, and Berlekamp and Wolfe subsequently developed an endgame and ''temperature'' theory for it (see references).  Armed with this they were able to construct plausible Go endgame positions from which they could give expert Go players a choice of sides and then defeat them either way.

Another game studied in the context of combinatorial game theory is [[chess]]. In 1953 [[Alan Turing]] wrote of the game, "If one can explain quite unambiguously in English, with the aid of mathematical symbols if required, how a calculation is to be done, then it is always possible to programme any digital computer to do that calculation, provided the storage capacity is adequate."<ref>{{cite web
 | url=https://turingarchive.kings.cam.ac.uk/publications-lectures-and-talks-amtb/amt-b-7 | title=Digital computers applied to games
 | author=Alan Turing
 | publisher=University of Southampton and King's College Cambridge
 | page=2
 }}</ref> In a 1950 paper, [[Claude Shannon]] estimated the lower bound of the [[game complexity|game-tree complexity]] of chess to be 10<sup>120</sup>, and today this is referred to as the [[Shannon number]].<ref>{{cite journal | author = Claude Shannon | author-link = Claude Shannon | title = Programming a Computer for Playing Chess | journal = Philosophical Magazine | volume = 41 | issue = 314 | year = 1950 | page = 4 | url = http://archive.computerhistory.org/projects/chess/related_materials/text/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon.062303002.pdf | url-status = dead | archiveurl = https://web.archive.org/web/20100706211229/http://archive.computerhistory.org/projects/chess/related_materials/text/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon.062303002.pdf | archivedate = 2010-07-06 }}</ref> Chess remains unsolved, although extensive study, including work involving the use of supercomputers has created chess endgame [[Endgame tablebase|tablebases]], which shows the result of perfect play for all end-games with seven pieces or less. [[Infinite chess]] has an even greater combinatorial complexity than chess (unless only limited end-games, or composed positions with a small number of pieces are being studied).