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Published online by Cambridge University Press: 29 May 2025
ABSTRACT. The traditional children's pencil-and-paper game called Dotsand- 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, where c is the total number of boxes in the game.
In this paper, we weaken this condition from to, and show this bound to be best possible.
Introduction to Loony Endgames and Controlled Value
The reader is assumed to be familiar with the game of Dots and Boxes. An excellent introduction can be found in [WW], [Nowakowski] or [D&B].
Some results about this game are more easily described in terms of the dual game, called Strings-and-Coins [D&B, Chapter 2]. This game is played on a graph G, whose nodes are called coins and whose branches are called strings. In this graph, the ends of each string are attached to two different coins or to a coin and the ground, which is a special uncapturable node. The valence of any coin is the number of strings attached to it. It has long been known that typical endgames reach a loony stage in which no coin has valence 1, and every coin of valence 2 is part of a chain of at least 3 such coins or of a loop of at least 4 such coins. When such a position occurs, the player who has just completed his turn is said to be in control. Let's call him Right. All moves now available to his opponent, called Left, are of the type called loony.
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