[236.—THE HAT PUZZLE.—solution]
I suggested that the reader should try this puzzle with counters, so I give my solution in that form. The silk hats are represented by black counters and the felt hats by white counters. The first row shows the hats in their original positions, and then each successive row shows how they appear after one of the five manipulations. It will thus be seen that we first move hats 2 and 3, then 7 and 8, then 4 and 5, then 10 and 11, and, finally, 1 and 2, leaving the four silk hats together, the four felt hats together, and the two vacant pegs at one end of the row. The first three pairs moved are dissimilar hats, the last two pairs being similar. There are other ways of solving the puzzle.
[237.—BOYS AND GIRLS.—solution]
There are a good many different solutions to this puzzle. Any contiguous pair, except 7-8, may be moved first, and after the first move there are variations. The following solution shows the position from the start right through each successive move to the end:—
| . | . | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 4 | 3 | 1 | 2 | . | . | 5 | 6 | 7 | 8 |
| 4 | 3 | 1 | 2 | 7 | 6 | 5 | . | . | 8 |
| 4 | 3 | 1 | 2 | 7 | . | . | 5 | 6 | 8 |
| 4 | . | . | 2 | 7 | 1 | 3 | 5 | 6 | 8 |
| 4 | 8 | 6 | 2 | 7 | 1 | 3 | 5 | . | . |
[238.—ARRANGING THE JAMPOTS.—solution]
Two of the pots, 13 and 19, were in their proper places. As every interchange may result in a pot being put in its place, it is clear that twenty-two interchanges will get them all in order. But this number of moves is not the fewest possible, the correct answer being seventeen. Exchange the following pairs: (3-1, 2-3), (15-4, 16-15), (17-7, 20-17), (24-10, 11-24, 12-11), (8-5, 6-8, 21-6, 23-21, 22-23, 14-22, 9-14, 18-9). When you have made the interchanges within any pair of brackets, all numbers within those brackets are in their places. There are five pairs of brackets, and 5 from 22 gives the number of changes required—17.