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.
The cooled game
Gt
G
t
G
t
Gt=\begin{cases}\{
L | |
G | |
t |
-t\mid
R | |
G | |
t |
+t\}&forallnumberst\leqanynumber\tauforwhichG\tauisinfinitesimallyclosetosomenumberm,\\ m&fort>\tau \end{cases}
t
G
\tau
G\tau
m
t(G)
G
G
G\tau
m
G
Heating is the inverse of cooling and is defined as the "integral"[6]
\inttG=\begin{cases}G&ifGisanumber,\\ \{\intt(GL)+t\mid\intt(GR)-t\}&otherwise. \end{cases}
Norton multiplication is an extension of multiplication to a game
G
U
G.U=\begin{cases}G x U&(i.e.thesumofGcopiesofU)ifGisanon-negativeinteger,\\ -G x -U&ifGisanegativeinteger,\\ \{GL.U+(U+I)\midGR.U-(U+I)\}whereIrangesover\Delta(U)&otherwise. \end{cases}
\Delta(U)
U
\{u-U:u\inUL\}\cup\{U-u:u\inUR\}
Overheating is an extension of heating used in Berlekamp's solution of Blockbusting,where
G
s
t
G,s,t
s>0
t | |
\int | |
s |
G=\begin{cases}G.s&ifGisaninteger,\\ \{
t | |
\int | |
s |
(GL)+t\mid
t | |
\int | |
s |
(GR)-t\}&otherwise. \end{cases}
Winning Ways also defines overheating of a game
G
X
t | |
\int | |
0 |
G=\left\{
t | |
\int | |
0 |
(GL)+X\mid
t | |
\int | |
0 |
(GR)-X\right\}
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
Chilling is a variant of cooling by
1
f(G)=\begin{cases}m&ifGisoftheformmorm*,\\ \{f(GL)-1\midf(GR)+1\}&otherwise. \end{cases}
1
G
Warming is a special case of overheating, namely
1 | |
\int | |
1* |
\int
G
\intG=\begin{cases}G&ifGisaneveninteger,\\ G*&ifGisanoddinteger,\\ \{\int(GL)+1\mid\int(GR)-1\}&otherwise. \end{cases}
. Elwyn Berlekamp . John H. . Conway . John Horton Conway . Richard K. . Guy . Richard K. Guy . Winning Ways for Your Mathematical Plays . Academic Press . 1982 . 978-0-12-091101-1 . 147, 163, 170 . Winning Ways for Your Mathematical Plays .