276.—­THE EIGHT VILLAS.

In one of the outlying suburbs of London a man had a square plot of ground on which he decided to build eight villas, as shown in the illustration, with a common recreation ground in the middle.  After the houses were completed, and all or some of them let, he discovered that the number of occupants in the three houses forming a side of the square was in every case nine.  He did not state how the occupants were distributed, but I have shown by the numbers on the sides of the houses one way in which it might have happened.  The puzzle is to discover the total number of ways in which all or any of the houses might be occupied, so that there should be nine persons on each side.  In order that there may be no misunderstanding, I will explain that although B is what we call a reflection of A, these would count as two different arrangements, while C, if it is turned round, will give four arrangements; and if turned round in front of a mirror, four other arrangements.  All eight must be counted.

[Illustration: 

/\      /\      /\
|2 |    |5 |    |2 |

/\              /\
|5 |            |5 |

/\      /\      /\
|2 |    |5 |    |2 |

+—–­+—–­+—–­+    +—–­+—–­+—–­+    +—–­+—–­+—–­+
| 1 | 6 | 2 |    | 2 | 6 | 1 |    | 1 | 6 | 2 |
+—–­+—–­+—–­+    +—–­+—–­+—–­+    +—–­+—–­+—–­+
| 6 |   | 6 |    | 6 |   | 6 |    | 4 |   | 4 |
+—–­+—–­+—–­+    +—–­+—–­+—–­+    +—–­+—–­+—–­+
| 2 | 6 | 1 |    | 1 | 6 | 2 |    | 4 | 2 | 3 |
+—–­+—–­+—–­+    +—–­+—–­+—–­+    +—–­+—–­+—–­+
A                B                C

]

277.—­COUNTER CROSSES.

All that we need for this puzzle is nine counters, numbered 1, 2, 3, 4, 5, 6, 7, 8, and 9.  It will be seen that in the illustration A these are arranged so as to form a Greek cross, while in the case of B they form a Latin cross.  In both cases the reader will find that the sum of the numbers in the upright of the cross is the same as the sum of the numbers in the horizontal arm.  It is quite easy to hit on such an arrangement by trial, but the problem is to discover in exactly how many different ways it may be done in each case.  Remember that reversals and reflections do not count as different.  That is to say, if you turn this page round you get four arrangements of the Greek cross, and if you turn it round again in front of a mirror you will get four more.  But these eight are all regarded as one and the same.  Now, how many different ways are there in each case?

[Illustration: 

(1) (2)

(2) (4) (5) (1) (6) (7)

(3) (4) (9) (5) (6) (3)

(7) (8)

A (8) B (9)

]

278.—­A DORMITORY PUZZLE.