HMMT 二月 2003 · GEN2 赛 · 第 6 题

HMMT February 2003 — GEN2 Round — Problem 6

The numbers 112, 121, 123, 153, 243, 313, and 322 are among the rows, columns, and diagonals of a 3 × 3 square grid of digits (rows and diagonals read left-to-right, and columns read top-to-bottom). What 3-digit number completes the list?

解析

The numbers 112, 121, 123, 153, 243, 313, and 322 are among the rows, columns, and diagonals of a 3 × 3 square grid of digits (rows and diagonals read left-to-right, and columns read top-to-bottom). What 3-digit number completes the list? Solution: 524 1 1 2 5 2 4 3 1 3 The center digit is the middle digit of 4 numbers (hence at least 3 members of the above list), so it must be 2. The top-left digit begins at least 2 members of the above list, so it must be 1 or 3. If it is 3, then after placing 313 we see that we need three more numbers starting with 3, impossible; hence, it is 1. So 243 and 313 must (in some order) be the last row and the last column, and now it is easy to complete the grid as shown; the answer is 524. 2