AMERICAN PUZZLES "15" AND "34,"
which have been christened "Boss." The materials of the puzzles are very simple, a description that may indeed be applied to all the amusements dealt with in this section. The puzzles, as purchased, consist of a square box of sixteen small wooden cubes, numbered from 1 to 16. The box of cubes may be purchased in the streets for a very trifling sum, or it may be obtained in the toy-shops in a more elaborate form, but still at a small cost. The popularity of the game may be guessed from the statement made by a New York toy-dealer to the effect that in one day he disposed of no less than 230 gross of a cheap variety. In London, street toy-vendors by the score sold them all day long for weeks together when they were first introduced, and a leading toy-dealer in the fashionable neighbourhood of Regent Street says the number sold retail from his shop daily was enormous. Their popularity in other countries is equally great.
The puzzle is twofold, and is described in the following quaint and curt manner in the little boxes sold in the streets:—
The Puzzle of Fifteen.—"Remove the 16 block. Put the pieces in the box irregularly, and arrange them to regular order by shoving."
The Magic Sixteen, or the Puzzle of Thirty-four.—"Arrange the sixteen blocks so that the sum of the numbers added up in any straight line, either vertical, horizontal, or diagonal, will be 34."
| 1 | 14 | 15 | 4 |
| 8 | 11 | 10 | 5 |
| 12 | 7 | 6 | 9 |
| 13 | 2 | 3 | 16 |
Fig. 1.—A Solution of the "34" Puzzle.
| 1 | 15 | 14 | 4 |
| 12 | 6 | 7 | 9 |
| 8 | 10 | 11 | 5 |
| 13 | 3 | 2 | 16 |
Fig. 2.—Another Solution of the "34" Puzzle.
It would appear that the "15" puzzle has the merit of being entirely new, a claim to which the "34" puzzle has no sort of right, it being found in many books of old and recent date. It is believed that there are in all sixteen different ways of arranging the numbered blocks so that the sum of the numbers will be 34 in every direction; but two ways will suffice to quote here, and they are as shown in Figs. 1, 2. The fascination and popularity of "Boss," however, all centre around the "15" puzzle; it is the solution of that which is said to have sent some people mad, to have made more forsake their ordinary occupation, and which claims to have given to a still larger and ever growing number of human beings a new incentive to life. The puzzle is fairly stated above in the words, "Put the pieces in the box irregularly," &c. As a first attempt, however, place the pieces as arranged when the "34" puzzle has been solved, and the "15" puzzle may be easily accomplished after a little practice. To describe the various moves would be unnecessary, but the object first to be aimed at is to get the first row of cubes, viz., 1, 2, 3, 4, into their proper places, attention being next directed to getting the 12 cube into its place; that cube will have to be again moved before all the cubes have been consecutively arranged, but it should always be kept as near to its proper position as possible. The cubes, when arranged, should read as follows (Fig. 3):—
| 1 | 2 | 3 | 4 |
| 5 | 6 | 7 | 8 |
| 9 | 10 | 11 | 12 |
| 13 | 14 | 15 |
Fig. 3.—"15" Puzzle—The Cubes in Order.
| 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.