HMMT 十一月 2015 · THM 赛 · 第 9 题

HMMT November 2015 — THM Round — Problem 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?

解析

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 x x 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 contain a 1, so one of the cells marked with a * must be a 1. Similarly, looking at the top left 4 × 3: 1 x x x x x x x x x * * One of the cells marked with a * must also contain a 1. But then the 3 × 3 square diagonally below the top left one: 1 x x x x x x * x x x * x * * ? must contain multiple 1s, which is a contradiction. Hence no such supersudokus exist.