By Michael H. Albert, Richard J. Nowakowski
This attention-grabbing examine combinatorial video games, that's, video games now not concerning probability or hidden details, bargains updates on typical video games similar to cross and Hex, on neutral video games resembling Chomp and Wythoff's Nim, and on elements of video games with infinitesimal values, plus analyses of the complexity of a few video games and puzzles and surveys on algorithmic online game thought, on taking part in to lose, and on dealing with cycles. the amount is rounded out with an up to date bibliography by means of Fraenkel and, for readers wanting to get their fingers soiled, a listing of unsolved difficulties by way of man and Nowakowski. Highlights contain a few of Siegel's groundbreaking paintings on crazy video games, the disclosing by way of Friedman and Landsberg of using renormalization to provide very fascinating effects approximately Chomp, and Nakamura's "Counting Liberties in taking pictures Races of Go." Like its predecessors, this e-book can be at the shelf of all severe video games fans.
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Example text
Or PushPush-1 puzzle: Another variation, called P USH-X, disallows the move the robot to the X robot from revisiting a square (the robot’s path by pushing light blocks. cannot cross). This direction was suggested in [DDO00a] because it immediately places the puzzles in NP. Demaine and Hoffmann [DH01] proved that P USH-1X and P USH P USH-1X are NP-complete. Hoff- ALGORITHMIC COMBINATORIAL GAME THEORY 35 mann’s reduction for P USH-* also establishes NP-completeness of P USH-*X without fixed blocks.
Lunar Lockout is another token-sliding puzzle, similar to Atomix in that the tokens slide until stopped. Lunar Lockout was produced by ThinkFun at one time; essentially the same game is now sold as “Pete’s Pike”. ) In Lunar Lockout there are no walls or barriers; a token may only slide if there is another token in place that will stop it. The goal is to get a particular token to a particular place. Thus, the rules are fairly simple and natural; however, the complexity is open, though there are partial results.
A secondary goal is to maximize the score, typically defined by k 2 points being awarded for removal of a group of k blocks. Biedl et al. [BDDC 02] proved that it is NP-complete to decide whether all blocks can be removed in a Clickomania puzzle. This complexity result holds even for puzzles with two columns and five colors, and for puzzles with five columns and three colors. On the other hand, for puzzles with one column (or, equivalently, one row) and arbitrarily many colors, they show that the maximum number of blocks can be removed in polynomial time.