Celebratio Mathematica — Berlekamp

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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