| 1 | 2 | 3 | 4 |
| 5 | 6 | 7 | 8 |
| 9 | 10 | 11 | 12 |
| 13 | 15 | 14 |
Fig. 4.—"15" Puzzle—The Cubes set for Solution.
"Boss," or the real American puzzle of "15" is to place the numbered cubes, as shown in Fig. 4, in the box, and then to arrange them, by sliding and without lifting any one cube, so that they shall read consecutively. It may at once be said that the American puzzle has never yet been solved. But why? is asked by every one, and every one tries to solve it. Articles on the puzzle have appeared in many periodicals, but no one has had the hardihood to publish a solution of the American puzzle. An ingenious calculator has stated that the fifteen cubes may be arranged in the box in 1,307,674,318,000 different combinations, and that it would take one individual a whole year to work out 105,000 of these arrangements, if only one arrangement was worked out every five minutes. Let the reader calculate at what remote period the whole of the different orders could be tested to see whether the "15-14" combination could be overcome. It seems to have been decided that there are a certain number of the combinations that can be solved, and that there are a certain number that cannot, and that the number of each is equal. If, when the fifteen cubes are placed in the box, the number of transpositions required to place the cubes in proper consecutive order is even, the puzzle may be solved; but if the number of transpositions required is odd, the puzzle cannot be solved. For example: take the first solution of the "34" puzzle (Fig. 1), and it will be found that six transpositions are required to place the numbers in the proper order, viz.:—
| 1. | Transpose | 14 | and | 2 |
| 2. | „ | 15 | „ | 3 |
| 3. | „ | 8 | „ | 5 |
| 4. | „ | 11 | „ | 6 |
| 5. | „ | 10 | „ | 7 |
| 6. | „ | 12 | „ | 9 |
The number of transpositions being even, the puzzle is soluble; with the "15-14" order, there being only one transposition necessary, or an odd number, the puzzle is insoluble. With this information and a little practice any player may tell at a glance when any combination of the figures is shown whether the puzzle is soluble or no.
After the above lengthy dissertation on these clever puzzles we will now proceed to minor topics which may be treated as arithmetical amusements.
| 2 | 9 | 4 |
| 7 | 5 | 3 |
| 6 | 1 | 8 |
Fig. 5.—The Magic Nine.
THE MAGIC NINE, OR THE PUZZLE OF FIFTEEN.
To arrange the numbers 1 to 9 in three rows, so that the sum of each row added together horizontally, vertically, or diagonally shall be 15. Fig. 5 shows how the arrangement has to be made.