1992 AIME Problems/Problem 12

Revision as of 13:40, 11 February 2008 by JBL (talk | contribs) (remarks on problem (still needs solution))

Problem

In a game of Chomp, two players alternately take bites from a 5-by-7 grid of unit squares. To take a bite, a player chooses one of the remaining squares, then removes ("eats") all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightward) of the chosen square. For example, the bite determined by the shaded square in the diagram would remove the shaded square and the four squares marked by $\times.$ (The squares with two or more dotted edges have been removed form the original board in previous moves.)

AIME 1992 Problem 12.png

The object of the game is to make one's opponent take the last bite. The diagram shows one of the many subsets of the set of 35 unit squares that can occur during the game of Chomp. How many different subsets are there in all? Include the full board and empty board in your count.

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it.

Links to Branches of Mathematics

This question is one case of the problem of counting order ideals or antichains in posets. Specifically, it ask for the number of order ideals of the product poset of the chain of length $4$ and the chain of length $6$.

See also

1992 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions
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