Cooling and heating (combinatorial game theory)

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In combinatorial game theory, cooling, heating, and overheating are operations on hot games to make them more amenable to the traditional methods of the theory, which was originally devised for cold games in which the winner is the last player to have a legal move.[1] Overheating was generalised by Elwyn Berlekamp for the analysis of Blockbusting.[2] Chilling (or unheating) and warming are variants used in the analysis of the endgame of Go.[3][4]

Cooling and chilling may be thought of as a tax on the player who moves, making them pay for the privilege of doing so, while heating, warming and overheating are operations that more or less reverse cooling and chilling.

Basic operations: cooling, heating[edit]

The cooled game (" cooled by ") for a game and a (surreal) number is defined by[5]


The amount by which is cooled is known as the temperature; the minimum for which is infinitesimally close to is known as the temperature of ; is said to freeze to ; is the mean value (or simply mean) of .

Heating is the inverse of cooling and is defined as the "integral"[6]

Multiplication and overheating[edit]

Norton multiplication is an extension of multiplication to a game and a positive game (the "unit") defined by[7]

The incentives of a game are defined as .

Overheating is an extension of heating used in Berlekamp's solution of Blockbusting, where overheated from to is defined for arbitrary games with as[8]

Winning Ways also defines overheating of a game by a positive game , as[9]

Note that in this definition numbers are not treated differently from arbitrary games.
Note that the "lower bound" 0 distinguishes this from the previous definition by Berlekamp

Operations for Go: chilling and warming [edit]

Chilling is a variant of cooling by used to analyse the Go endgame of Go and is defined by[10]

This is equivalent to cooling by when is an "even elementary Go position in canonical form".[11]

Warming is a special case of overheating, namely , normally written simply as which inverts chilling when is an "even elementary Go position in canonical form". In this case the previous definition simplifies to the form[12]


  1. ^ Berlekamp, Elwyn R.; Conway, John H.; Guy, Richard K. (1982). Winning Ways for Your Mathematical Plays. Academic Press. pp. 147, 163, 170. ISBN 978-0-12-091101-1.
  2. ^ Berlekamp, Elwyn (January 13, 1987). "Blockbusting and Domineering". Journal of Combinatorial Theory (published September 1988). 49 (1): 67–116. doi:10.1016/0097-3165(88)90028-3.[permanent dead link]
  3. ^ Berlekamp, Elwyn; Wolfe, David (1997). Mathematical Go: Chilling Gets the Last Point. A K Peters Ltd. ISBN 978-1-56881-032-4.
  4. ^ Berlekamp, Elwyn; Wolfe, David (1994). Mathematical Go Endgames. Ishi Press. pp. 50–55. ISBN 978-0-923891-36-7. (paperback version of Mathematical Go: Chilling Gets the Last Point)
  5. ^ Berlekamp, Conway & Guy (1982), p. 147
  6. ^ Berlekamp, Conway & Guy (1982), p. 163
  7. ^ Berlekamp, Conway & Guy (1982), p. 246
  8. ^ Berlekamp (1987), p. 77
  9. ^ Berlekamp, Conway & Guy (1982), p. 170
  10. ^ Berlekamp & Wolfe (1994), p. 53
  11. ^ Berlekamp & Wolfe (1994), pp. 53–55
  12. ^ Berlekamp & Wolfe (1994), pp. 52–55