HMMT 二月 2008 · COMB 赛 · 第 6 题

英文解析

6. [ 5 ] A Sudoku matrix is defined as a 9 × 9 array with entries from { 1 , 2 , . . . , 9 }
   and with the constraint that each row, each column, and each of the nine 1
   3 × 3 boxes that tile the array contains each digit from 1 to 9 exactly once.2
   ?
   A Sudoku matrix is chosen at random (so that every Sudoku matrix hasequal probability of being chosen). We know two of squares in this matrix,
   as shown. What is the probability that the square marked by ? containsthe digit 3?
   Answer: The third row must contain the digit 1, and it cannot appear in the leftmost three 2
   squares. Therefore, the digit 1 must fall into one of the six squares shown below that are marked with ? .21
   By symmetry, each starred square has an equal probability of containing the digit 1 (To see this moreprecisely, note that swapping columns 4 and 5 gives another Sudoku matrix, so the probability that the 4 th column ? has the 1 is the same as the probability that the 5 th column ? has the 1. Similarly,
   switching the 4-5-6 th columns with the 7-8-9 th columns yields another Sudoku matrix, which impliesin particular that the probability that the 4 th column ? has the 1 is the same as the probability that the 7 th column ? has the 1. The rest of the argument follows analogously.) Therefore, the probabilitythat the ? square contains 1 is 1 / 6.
   ? ? ? ? ? ?21
   Similarly the probability that the digit 2 appears at ? is also 1 / 6. By symmetry, the square ? has equal
   ( )
   1 1
   probability of containing the digits 3 , 4 , 5 , 6 , 7 , 8 , 9. It follows that this probability is 1 − − / 7 =
   6 6
   .2
   21