HMMT 十一月 2013 · 冲刺赛 · 第 11 题

HMMT November 2013 — Guts Round — Problem 11

[ 8 ] David has a unit triangular array of 10 points, 4 on each side. A looping path is a sequence A , A , . . . , A containing each of the 10 points exactly once, such that A and A are adjacent 1 2 10 i i +1 (exactly 1 unit apart) for i = 1 , 2 , . . . , 10. (Here A = A .) Find the number of looping paths in this 11 1 array. 2 2 2 2

解析

[ 8 ] David has a unit triangular array of 10 points, 4 on each side. A looping path is a sequence A , A , . . . , A containing each of the 10 points exactly once, such that A and A are adjacent 1 2 10 i i +1 (exactly 1 unit apart) for i = 1 , 2 , . . . , 10. (Here A = A .) Find the number of looping paths in this 11 1 array. Answer: 60 There are 10 · 2 times as many loop sequences as loops. To count the number of loops, first focus on the three corners of the array: their edges are uniquely determined. It’s now easy to see there are 3 loops (they form “ V -shapes”), so the answer is 10 · 2 · 3 = 60. 2 2 2 2