HMMT 二月 2000 · 几何 · 第 1 题
HMMT February 2000 — Geometry — Problem 1
题目详情
- Ho w man y re tangles are there on an 8 x 8 he k erb oard?
解析
- Consider the b oard lab eled as b elo w, with lab els for olumns and ro ws. T o ho ose an y re tangle on the b oard, it is suÆ ien t to ho ose some n um b er (1-8) of adja en t olumns, and some n um b er (1-8) of adja en t ro ws, sin e the re tangle an b e reated b y forming the in terse tion of the olumns and ro ws. F or instan e, the in terse tion of olumns 2,3 and ro ws 3,4,5 is the re tangle shaded b elo w. So, there are 8 w a ys to ho ose 1 adja en t olumn, 7 w a ys to ho ose 2 adja en t olumns, : : : , 1 w a y to ho ose 8 adja en t olumns, so there are 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 36 total w a ys to ho ose the olumns, and 36 w a ys to ho ose the ro ws. Th us, the total n um b er of w a ys to ho ose a re tangle (i.e. 2 the total n um b er of re tangles) is 36 = 1296 . 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8