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HMMT 二月 2001 · 冲刺赛 · 第 46 题

HMMT February 2001 — Guts Round — Problem 46

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ ± 5] Draw a rectangle. Connect the midpoints of the opposite sides to get 4 congruent rectangles. Connect the midpoints of the lower right rectangle for a total of 7 rectangles. Repeat this process infinitely. Let n be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing an edge have the same color and m be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing a corner have the same color. Find the ordered pair ( n, m ).
解析
  1. [ ± 5] Draw a rectangle. Connect the midpoints of the opposite sides to get 4 congruent rectangles. Connect the midpoints of the lower right rectangle for a total of 7 rectangles. Repeat this process infinitely. Let n be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing an edge have the same color and m be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing a corner have the same color. Find the ordered pair ( n, m ). Solution: (3 , 4) .