chomp

WordNet

chew noisily; "The boy chomped his sandwich" (同)champ
the act of chewing noisily

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/04/07 16:38:50」(JST)

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A chocolate bar divided into a 5 × 3 grid of squares, ready for a game of Chomp

Chomp is a two-player strategy game played on a rectangular chocolate bar made up of smaller square blocks (cells). The players take it in turns to choose one block and "eat it" (remove from the board), together with those that are below it and to its right. The top left block is "poisoned" and the player who eats this loses.

The chocolate-bar formulation of Chomp is due to David Gale, but an equivalent game expressed in terms of choosing divisors of a fixed integer was published earlier by Frederik Schuh.

Chomp is a special case of a poset game where the partially ordered set on which the game is played is a product of total orders with the minimal element (poisonous block) removed.

Example game

Below shows the sequence of moves in a typical game starting with a 3 × 5 bar:

Initially		Player A		Player B		Player A		Player B

Player A must eat the last block and so loses. Note that since it is provable that player A can win, at least one of A's moves is a mistake.

Who wins?

Chomp belongs to the category of impartial two-player perfect information games.

It turns out that for any rectangular starting position other than 1×1 the first player can win. This can be shown using a strategy-stealing argument: assume that the second player has a winning strategy against any initial first-player move. Suppose then, that the first player takes only the bottom right hand square. By our assumption, the second player has a response to this which will force victory. But if such a winning response exists, the first player could have played it as his first move and thus forced victory. The second player therefore cannot have a winning strategy.

Computers can easily calculate winning moves for this game on two-dimensional boards of reasonable size.

Generalisations of Chomp

Three-dimensional Chomp has an initial chocolate bar of a cuboid of blocks indexed as (i,j,k). A move is to take a block together with any block all of whose indices are greater or equal to the corresponding index of the chosen block. In the same way Chomp can be generalised to any number of dimensions.

Chomp is sometimes described numerically. An initial natural number is given, and players alternate choosing positive divisors of the initial number, but may not choose 1 or a multiple of a previously chosen divisor. This game models n-dimensional Chomp, where the initial natural number has n prime factors and the dimensions of the Chomp board are given by the exponents of the primes in its prime factorization. Ordinal Chomp is played on an infinite board with some of its dimensions ordinal numbers: for example a 2 × (ω + 4) bar. A move is to pick any block and remove all blocks with both indices greater than or equal the corresponding indices of the chosen block. The case of ω × ω × ω Chomp is a notable open problem; a $100 reward has been offered^[1] for finding a winning first move.

More generally, Chomp can be played on any partially ordered set with a least element. A move is to remove any element along with all larger elements. A player loses by taking the least element.

All varieties of Chomp can also be played without resorting to poison by using the misère play convention: The player who eats the final chocolate block is not poisoned, but simply loses by virtue of being the last player. This is identical to the ordinary rule when playing Chomp on its own, but differs when playing the disjunctive sum of Chomp games, where only the last final chocolate block loses.

References

^ p. 482 in: Games of No Chance (R. J. Nowakowski, ed.), Cambridge University Press, 1998.

External links

More information about the game
A freeware version for windows
Play Chomp online

English Journal

CHoMP: a chemoenzymatic histology method using clickable probes.

Rouhanifard SH1, López-Aguilar A, Wu P.
Chembiochem : a European journal of chemical biology.Chembiochem.2014 Dec 15;15(18):2667-73. doi: 10.1002/cbic.201402433. Epub 2014 Nov 17.
The characterization of aberrant glycosylation patterns in biopsied patient samples represents a remarkable challenge for scientists and medical doctors due to the lack of specific methods for detection. Here, we report the development of a histological method, dubbed CHoMP-chemoenzymatic histology
PMID 25403986

Three-dimensional computer simulations of feeding behaviour in red and giant pandas relate skull biomechanics with dietary niche partitioning.

Figueirido B1, Tseng ZJ, Serrano-Alarcón FJ, Martín-Serra A, Pastor JF.
Biology letters.Biol Lett.2014 Apr 9;10(4):20140196. doi: 10.1098/rsbl.2014.0196. Print 2014.
The red (Ailurus fulgens) and giant (Ailuropoda melanoleuca) pandas are mammalian carnivores convergently adapted to a bamboo feeding diet. However, whereas Ailurus forages almost entirely on younger leaves, fruits and tender trunks, Ailuropoda relies more on trunks and stems. Such difference in for
PMID 24718096

Process and impact evaluation of the Romp & Chomp obesity prevention intervention in early childhood settings: lessons learned from implementation in preschools and long day care settings.

de Silva-Sanigorski AM1, Bell AC, Kremer P, Park J, Demajo L, Smith M, Sharp S, Nichols M, Carpenter L, Boak R, Swinburn B.
Childhood obesity (Print).Child Obes.2012 Jun;8(3):205-15. doi: 10.1089/chi.2011.0118.
BACKGROUND: The Romp & Chomp controlled trial, which aimed to prevent obesity in preschool Australian children, was recently found to reduce the prevalence of childhood overweight and obesity and improve children's dietary patterns. The intervention focused on capacity building and policy implem
PMID 22799546

Japanese Journal

A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems

Arai Zin,Kalies William,Kokubu Hiroshi,Mischaikow Konstantin,Oka Hiroe,Pilarczyk Paweł
SIAM Journal on Applied Dynamical Systems 8(3), 757-789, 2009-07-02
… An interactive visualization of the results of computations discussed in the paper can be accessed at the Web site http://chomp.rutgers.edu/database/, and the source code of the software used to obtain these results has also been made freely available. …
NAID 120001629760

CHomPソフトウェア入門

平岡裕章,ピラルチックパヴェル
応用数理 18(1), 48-55, 2008-03-25
NAID 110006633787

毒まみれ半順序付き集合ゲームの必勝法(ネットワークプロセッサ,通信のための信号処理,符号理論,一般)

高田智史,伊藤大雄,中村義作
電子情報通信学会技術研究報告. CS, 通信方式 105(638), 13-18, 2006-02-28
… Poset Gameは有名なNimから未解決のChompまで多様なゲームを含み、この研究から様々な面白い結果が得られている。 …
NAID 110004735370

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