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## THE EIGHT VILLAS.

(

Combination and Group Problems)

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

]

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