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By Edwin Beckenbach

Collage of CaliforniaEngineering and actual sciences extension series.Includes bibliographies.Based at the Statewide lecture sequence on combinatorial arithmetic provided by means of the collage of California, collage Extension, Engineering and actual Sciences department, in 1962.

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That maximizing the number of moves in a single component is NP-complete (for n  n boards). In a general endgame, deciding the outcome may not be in NP because it is difficult to prove that the opponent has no better strategy. However, Buro [Bur00] proved that this problem is NP-equivalent [GJ79], that is, the problem can be solved by a polynomial number of calls to an algorithm for any NP-complete problem, and vice versa. 16), the complexity of deciding the outcome of a general Amazons position remained open for several years, only to be solved nearly simultaneously by multiple people.

12. Jigsaw, edge-matching, tiling, and packing puzzles. Jigsaw puzzles [Wil04] are another one of the most popular kinds of puzzles, dating back to the 1760s. One way to formalize such puzzles is as a collection of square pieces, where each side is either straight or augmented with a tab or a pocket of a particular shape. The goal is to arrange the given pieces so that they form exactly a given rectangular shape. Although this formalization does not explicitly allow for patterns on pieces to give hints about whether pieces match, this information can simply be encoded into the shapes of the tabs and pockets, making them compatible only when the patterns also match.

It remains open whether Dots-and-Boxes or Strings-and-Coins are in NP or PSPACE-complete from an arbitrary configuration. Even the case of a 1n grid of boxes is not fully understood from a Combinatorial Game Theory perspective [GN02]. 13. Amazons. Amazons is a game invented by Walter Zamkauskas in 1988, containing elements of Chess and Go. Gameplay takes place on a 10  10 board with four amazons of each color arranged as in Figure 7 (left). In each turn, Left [Right] moves a black [white] amazon to any unoccupied square accessible by a Chess queen’s move, and fires an arrow to any unoccupied square reachable by a Chess queen’s move from the amazon’s new position.

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