HMMT 十一月 2008 · 冲刺赛 · 第 14 题

HMMT November 2008 — Guts Round — Problem 14

[ 9 ] You have a 2 × 3 grid filled with integers between 1 and 9. The numbers in each row and column are distinct, the first row sums to 23, and the columns sum to 14 , 16, and 17 respectively. 14 16 17 23 a b c x y z What is x + 2 y + 3 z ?

解析

[ 9 ] You have a 2 × 3 grid filled with integers between 1 and 9. The numbers in each row and column are distinct, the first row sums to 23, and the columns sum to 14 , 16, and 17 respectively. 14 16 17 23 a b c x y z What is x + 2 y + 3 z ? Answer: 49 The sum of all 6 numbers is 14 + 16 + 17 = 47, so x + y + z = 47 − 23 = 24. If three distinct digits sum to 24, they must be 7, 8, and 9, because any other triple of digits would have a smaller sum. So, we try placing these digits in for x , y , and z , and the only arrangement that does not force equal digits in any row or column is x = 8 , y = 7 , z = 9. In this case, x + 2 y + 3 z = 49.