HMMT 二月 2003 · GEN2 赛 · 第 6 题
HMMT February 2003 — GEN2 Round — Problem 6
题目详情
英文原题
- The numbers 112, 121, 123, 153, 243, 313, and 322 are among the rows, columns, anddiagonals of a 3 × 3 square grid of digits (rows and diagonals read left-to-right, andcolumns read top-to-bottom). What 3-digit number completes the list?
解析
英文解析
- The numbers 112, 121, 123, 153, 243, 313, and 322 are among the rows, columns, anddiagonals of a 3 × 3 square grid of digits (rows and diagonals read left-to-right, andcolumns 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 theabove list), so it must be 2. The top-left digit begins at least 2 members of the abovelist, so it must be 1 or 3. If it is 3, then after placing 313 we see that we need threemore numbers starting with 3, impossible; hence, it is 1. So 243 and 313 must (in someorder) be the last row and the last column, and now it is easy to complete the grid asshown; the answer is 524. 2