HMMT 十一月 2015 · THM 赛 · 第 9 题

英文解析

9. Consider a 9 × 9 grid of squares. Haruki fills each square in this grid with an integer between 1 and 9,
   inclusive. The grid is called a super-sudoku if each of the following three conditions hold:
   • Each column in the grid contains each of the numbers 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 exactly once.
   • Each row in the grid contains each of the numbers 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 exactly once.
   • Each 3 × 3 subsquare in the grid contains each of the numbers 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 exactly once.
   How many possible super-sudoku grids are there?
   Proposed by: Alexander Katz
   Answer: 0
   Without loss of generality, suppose that the top left corner contains a 1, and examine the top left 3 × 4:
   1 x x xx x x *
   x x x *
   There cannot be another 1 in any of the cells marked with an x, but the 3 × 3 on the right must containa 1, so one of the cells marked with a * must be a 1. Similarly, looking at the top left 4 × 3:
   1 x xx x xx x xx * *
   One of the cells marked with a * must also contain a 1. But then the 3 × 3 square diagonally belowthe top left one:
   1 x x xx x x *
   x x x *
   x * * ?
   must contain multiple 1 s, which is a contradiction. Hence no such supersudokus exist.