Difference between revisions of "1992 AIME Problems/Problem 12"
m (→See also: box) |
(remarks on problem (still needs solution)) |
||
| Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
| − | In a game of ''Chomp'', two players alternately take bites from a 5-by-7 grid of unit | + | In a game of ''[[Chomp]]'', two players alternately take bites from a 5-by-7 grid of [[unit square]]s. To take a bite, a player chooses one of the remaining [[square (geometry) | 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 <math>\times.</math> (The squares with two or more dotted edges have been removed form the original board in previous moves.) |
[[Image:AIME_1992_Problem_12.png]] | [[Image: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 | + | The object of the game is to make one's opponent take the last bite. The diagram shows one of the many [[subset]]s 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 == | == Solution == | ||
{{solution}} | {{solution}} | ||
| + | |||
| + | == Links to Branches of Mathematics == | ||
| + | This question is one case of the problem of counting [[order ideal]]s or [[antichain]]s in [[poset]]s. Specifically, it ask for the number of order ideals of the [[product poset]] of the [[chain]] of length <math>4</math> and the chain of length <math>6</math>. | ||
== See also == | == See also == | ||
{{AIME box|year=1992|num-b=11|num-a=13}} | {{AIME box|year=1992|num-b=11|num-a=13}} | ||
Revision as of 12:40, 11 February 2008
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 
(The squares with two or more dotted edges have been removed form the original board in previous moves.)
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 
and the chain of length 
.
See also
| 1992 AIME (Problems • Answer Key • Resources) | ||
| 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 | ||
