401.—­EIGHT JOLLY GAOL-BIRDS.

There are eight ways of forming the magic square—­all merely different aspects of one fundamental arrangement.  Thus, if you give our first square a quarter turn you will get the second square; and as the four sides may be in turn brought to the top, there are four aspects.  These four in turn reflected in a mirror produce the remaining four aspects.  Now, of these eight arrangements only four can possibly be reached under the conditions, and only two of these four can be reached in the fewest possible moves, which is nineteen.  These two arrangements are shown.  Move the men in the following order:  5, 3, 2, 5, 7, 6, 4, 1, 5, 7, 6, 4, 1, 6, 4, 8, 3, 2, 7, and you get the first square.  Move them thus:  4, 1, 2, 4, 1, 6, 7, 1, 5, 8, 1, 5, 6, 7, 5, 6, 4, 2, 7, and you have the arrangement in the second square.  In the first case every man has moved, but in the second case the man numbered 3 has never left his cell.  Therefore No. 3 must be the obstinate prisoner, and the second square must be the required arrangement.

[Illustration: 

+—–­+—–­+—–­+     +—–­+—–­+—–­+
|   |   |   |     |   |   |   |
| 5       7 |     | 7   4   3 |
|   |   |   |     |   |   |   |
+- -+- -+- -+     +- -+- -+- -+
|   |   |   |     |   |   |   |
| 6   4   2 |     |     4   8 |
|   |   |   |     |   |   |   |
+- -+- -+- -+     +- -+- -+- -+
|   |   |   |     |   |   |   |
| 1   8   3 |     | 5   6   1 |
|   |   |   |     |   |   |   |
+—–­+—–­+—–­+     +—–­+—–­+—–­+

]

402.—­NINE JOLLY GAOL BIRDS.

There is a pitfall set for the unwary in this little puzzle.  At the start one man is allowed to be placed on the shoulders of another, so as to give always one empty cell to enable the prisoners to move about without any two ever being in a cell together.  The two united prisoners are allowed to add their numbers together, and are, of course, permitted to remain together at the completion of the magic square.  But they are obviously not compelled so to remain together, provided that one of the pair on his final move does not break the condition of entering a cell already occupied.  After the acute solver has noticed this point, it is for him to determine which method is the better one—­for the two to be together at the count or to separate.  As a matter of fact, the puzzle can be solved in seventeen moves if the men are to remain together; but if they separate at the end, they may actually save a move and perform the feat in sixteen!  The trick consists in placing the man in the centre on the back of one of the corner men, and then working the pair into the centre before their final separation.

[Illustration: 

A                 B
+—–­+—–­+—–­+     +—–­+—–­+—–­+
|   |   |   |     |   |   |   |
| 2   9   4 |     | 6   7   2 |
|   |   |   |     |   |   |   |
+- -+- -+- -+     +- -+- -+- -+
|   |   |   |     |   |   |   |
| 7   5   3 |     | 1   5   9 |
|   |   |   |     |   |   |   |
+- -+- -+- -+     +- -+- -+- -+
|   |   |   |     |   |   |   |
| 6   1   8 |     | 8   3   4 |
|   |   |   |     |   |   |   |
+—–­+—–­+—–­+     +—–­+—–­+—–­+

]