| 7,560 = | 23 | × | 33 | × | 5 | × | 7 | |
| Powers = | 3 | 3 | 1 | 1 | ||||
| Therefore | 4 | × | 4 | × | 2 | × | 2 | = 64 divisors. |
To find the smallest number that has a given number of divisors we must proceed by trial. But it is important sometimes to note whether or not the condition is that there shall be a given number of divisors and no more. For example, the smallest number that has seven divisors and no more is 64, while 24 has eight divisors, and might equally fulfil the conditions. The stipulation as to "no more" was not necessary in the case of my puzzle, for no smaller number has more than sixty-four divisors.
[13].—The Man of Law's Puzzle.
The fewest possible moves for getting the prisoners into their dungeons in the required numerical order are twenty-six. The men move in the following order:—1, 2, 3, 1, 2, 6, 5, 3, 1, 2, 6, 5, 3, 1, 2, 4, 8, 7, 1, 2, 4, 8, 7, 4, 5, 6. As there are never more than one vacant dungeon to be moved into, there can be no ambiguity in the notation.
The diagram may be simplified by my "buttons and string" method, fully explained in A. in M., p. 230. It then takes one of the simple forms of A or B, and the solution is much easier. In A we use counters; in B we can employ rooks on a corner of a chessboard. In both cases we have to get the order
in the fewest possible moves.
See also solution to No. [94].