HMMT 二月 2008 · 冲刺赛 · 第 3 题
HMMT February 2008 — Guts Round — Problem 3
题目详情
- [ 5 ] How many ways can you color the squares of a 2 × 2008 grid in 3 colors such that no two squaresof the same color share an edge?
11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUNDth 2
英文原题
[ 5 ] How many ways can you color the squares of a 2 × 2008 grid in 3 colors such that no two squares
of the same color share an edge?
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11 th HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUND
解析
英文解析
- [ 5 ] How many ways can you color the squares of a 2 × 2008 grid in 3 colors such that no two squaresof the same color share an edge?
2008
Answer: 2 · 3 Denote the colors A, B, C . The left-most column can be colored in 6 ways. Foreach subsequent column, if the k th column is colored with AB , then the ( k + 1)th column can only becolored with one of BA, BC, CA . That is, if we have colored the first k columns, then there are 3 ways
2007
to color the ( k + 1)th column. It follows that the number of ways of coloring the board is 6 × 3 .
11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUNDth 2