Celebratio Mathematica

E. Berlekamp: “The economist’s view of combinatorial games,” pp. 365–405 in More games of no chance (Berkeley, CA, 11–21 July 2000). Edited by R. J. Nowakowski. MSRI Publications 29. Cambridge University Press, 1996. MR 1427978 Zbl 0872.90131 incollection

We introduce two equivalent methodologies for defining and computing a position’s mean (value of playing Black rather than White) and temperature (value of next move). Both methodologies apply in more generality than the classical one. The first, following the notion of a free market, relies on the transfer of a “tax” between players, determined by continuous competitive auctions. The second relies on a generalized thermograph, which reduces to the classical thermograph when the game is loop-free.

When a sum of games is played optimally according the economic rules described, the mean (which is additive) and the temperature determine the final score precisely.

This framework extends and refines several classical notions. Thus, finite games that are numbers in Conway’s sense are now seen to have negative natural temperatures. All games can now be viewed as terminating naturally with integer scores when the temperature reaches \( -1 \).

E. Berlekamp and Y. Kim: “Where is the ‘thousand-dollar ko’?,” pp. 203–226 in More games of no chance (Berkeley, CA, 11–21 July 2000). Edited by R. J. Nowakowski. MSRI Publications 29. Cambridge University Press, 1996. An earlier version of this was published in Go World 71 (1994). MR 1427966 Zbl 0873.90141 incollection

Yonghoan Kim	Related
Richard Joseph Nowakowski	Related

E. Berlekamp: “The 4G4G4G4G4 problems and solutions,” pp. 231–241 in More games of no chance (Berkeley, CA, 24–28 July 2000). Edited by R. J. Nowakowski. MSRI Publications 42. Cambridge University Press, 2002. This is related to an article published in Puzzlers’ tribute: A feast for the mind (2002). MR 1973015 Zbl 1062.91522 incollection

E. Berlekamp and K. Scott: “Forcing your opponent to stay in control of a loony dots-and-boxes endgame,” pp. 317–330 in More games of no chance (Berkeley, CA, 24–28 July 2000). Edited by R. J. Nowakowski. MSRI Publications 42. Cambridge University Press, 2002. MR 1973020 Zbl 1062.91523 incollection

The traditional children’s pencil-and-paper game called Dots-and-Boxes is a contest to outscore the opponent by completing more boxes. It has long been known that winning strategies for certain types of positions in this game can be copied from the winning strategies for another game called Nimstring, which is played according to similar rules except that the Nimstring loser is whichever player completes the last box. Under certain common but restrictive conditions, one player (Right) achieves his optimal Dots-and-Boxes score, \( v \), by playing so as to win the Nimstring game. An easily computed lower bound on \( v \) is known as the controlled value, \( cv \). Previous results asserted that \( v = cv \) if \( cv \geq c/2 \), where \( c \) is the total number of boxes in the game.

In this paper, we weaken this condition from \( cv \geq c \) to \( cv \geq 10 \), and show this bound to be best possible.

Katherine Scott	Related
Richard Joseph Nowakowski	Related

E. Berlekamp: “Idempotents among partisan games,” pp. 3–23 in More games of no chance (Berkeley, CA, 24–28 July 2000). Edited by R. J. Nowakowski. MSRI Publications 42. Cambridge University Press, 2002. MR 1973000 Zbl 1047.91521 incollection

E. Berlekamp: “Yellow-brown Hackenbush,” pp. 413–418 in Games of no chance 3 (Banff, AB, June 2005). Edited by M. H. Albert and R. J. Nowakowski. MSRI Publications 56. Cambridge University Press, 2009. Zbl 1192.91048 incollection

Michael Henry Albert	Related
Richard Joseph Nowakowski	Related

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