The number of different ways in which eight persons, with eight hats, can each take the wrong hat, is 14,833.
Here are the successive solutions for any number of persons from one to eight:—
1 = 0
2 = 1
3 = 2
4 = 9
5 = 44
6 = 265
7 = 1,854
8 = 14,833
To get these numbers, multiply successively by 2, 3, 4, 5, etc. When the multiplier is even, add 1; when odd, deduct 1. Thus, 3 × 1 - 1 = 2, 4 × 2 + 1 = 9; 5 × 9 - 1 = 44; and so on. Or you can multiply the sum of the number of ways for n-1 and n-2 persons by n-1, and so get the solution for n persons. Thus, 4(2 + 9) = 44; 5(9 + 44) = 265; and so on.
[268.—THE PEAL OF BELLS.—solution]
The bells should be rung as follows:—
| 1 2 3 4 |
| 2 1 4 3 |
| 2 4 1 3 |
| 4 2 3 1 |
| 4 3 2 1 |
| 3 4 1 2 |
| 3 1 4 2 |
| 1 3 2 4 |
| 3 1 2 4 |
| 1 3 4 2 |
| 1 4 3 2 |
| 4 1 2 3 |
| 4 2 1 3 |
| 2 4 3 1 |
| 2 3 4 1 |
| 3 2 1 4 |
| 2 3 1 4 |
| 3 2 4 1 |
| 3 4 2 1 |
| 4 3 1 2 |
| 4 1 3 2 |
| 1 4 2 3 |
| 1 2 4 3 |
| 2 1 3 4 |
I have constructed peals for five and six bells respectively, and a solution is possible for any number of bells under the conditions previously stated.