| 107 | 134 | 235 |
| 249 | 586 | 746 |
| 356 | 720 | 981 |
Of course, in each case figures in the first two lines may be exchanged vertically without altering the total, and as a result there are just 3,072 different ways in which the figures might be actually placed on the locker doors. I must content myself with showing one little principle involved in this puzzle. The sum of the digits in the total is always governed by the digit omitted. 9/9 - 7/10 - 5/11 - 3/12 - 1/13 - 8/14 - 6/15 - 4/16 - 2/17 - 0/18. Whichever digit shown here in the upper line we omit, the sum of the digits in the total will be found beneath it. Thus in the case of locker A we omitted 8, and the figures in the total sum up to 14. If, therefore, we wanted to get 356, we may know at once to a certainty that it can only be obtained (if at all) by dropping the 8.
[80.—THE THREE GROUPS.—solution]
There are nine solutions to this puzzle, as follows, and no more:—
| 12 × 483 = 5,796 |
| 27 × 198 = 5,346 |
| 42 × 138 = 5,796 |
| 39 × 186 = 7,254 |
| 18 × 297 = 5,346 |
| 48 × 159 = 7,632 |
| 28 × 157 = 4,396 |
| 4 × 1,738 = 6,952 |
| 4 × 1,963 = 7,852 |
The seventh answer is the one that is most likely to be overlooked by solvers of the puzzle.
[81.—THE NINE COUNTERS.—solution]
In this case a certain amount of mere "trial" is unavoidable. But there are two kinds of "trials"—those that are purely haphazard, and those that are methodical. The true puzzle lover is never satisfied with mere haphazard trials. The reader will find that by just reversing the figures in 23 and 46 (making the multipliers 32 and 64) both products will be 5,056. This is an improvement, but it is not the correct answer. We can get as large a product as 5,568 if we multiply 174 by 32 and 96 by 58, but this solution is not to be found without the exercise of some judgment and patience.