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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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