There are several ways of solving the puzzle, but there is very little
difference between them. The solver should, however, first of all bear
in mind that in making his calculations he need only consider the four
villas that stand at the corners, because the intermediate villas can
never vary when the corners are known. One way is to place the numbers
nought to 9 one at a time in the top left-hand corner, and then consider
each case in turn.
Now, if we place 9 in the corner as shown in the Diagram A, two of the
corners cannot be occupied, while the corner that is diagonally opposite
may be filled by 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 persons. We thus see
that there are 10
[Illustration:
+---+---+ +-+-----+ +---+---+
|O OHO O| |OHO O O| |O OHO O|
| H | | + | | +=+ |
|O OHO O| |OHO O O| |O OHOHO|
+-+ +-+-+ +-+-----+ +---+ + |
|O|O O|O| |O|O O O| |O O O|O|
| +---+ | | +-+-+ | | +-+ |
|O O O O| |O O OHO| |O O|O O|
+-------+ +-------+ +-------+
2 3 4
+-----+-+ +-+-----+ +-------+
|O O OHO| |OHO O O| |O O O O|
| +=+ | | +=+ | | +=+=+=+
|O OHO O| |OHOHO O| |OHOHO O|
| +-+-+ + | + +-+ | + + + |
|O|O O|O| |O|O O|O| |O|OHO O|
+=+ +=+ | + +=+ +=+ + |
|O O O O| |OHO O O| |O O|O O|
+-------+ +-+-----+ +---+---+
5 6 7
THE SIXTEEN SHEEP
]
solutions with a 9 in the corner. If, however, we substitute 8, the two
corners in the same row and column may contain 0, 0, or 1, 1, or 0, 1,
or 1, 0. In the case of B, ten different selections may be made for the
fourth corner; but in each of the cases C, D, and E, only nine
selections are possible, because we cannot use the 9. Therefore with 8
in the top left-hand corner there are 10 + (3 × 9) = 37 different
solutions. If we then try 7 in the corner, the result will be 10 + 27 +
40, or 77 solutions. With 6 we get 10 + 27 + 40 + 49 = 126; with 5, 10 +
27 + 40 + 49 + 54 = 180; with 4, the same as with 5, + 55 = 235 ; with
3, the same as with 4, + 52 = 287; with 2, the same as with 3, + 45 =
332; with 1, the same as with 2, + 34 = 366, and with nought in the top
left-hand corner the number of solutions will be found to be 10 + 27 +
40 + 49 + 54 + 55 + 52 + 45 + 34 + 19 = 385. As there is no other number
to be placed in the top left-hand corner, we have now only to add these
totals together thus, 10 + 37 + 77 + 126 + 180 + 235 + 287 + 332 + 366 +
385 = 2,035. We therefore find that the total number of ways in which
tenants may occupy some or all of the eight villas so that there shall
be always nine persons living along each side of the square is 2,035. Of
course, this method must obviously cover all the reversals and
reflections, since each corner in turn is occupied by every number in
all possible combinations with the other two corners that are in line
with it.
[Illustration:
A B C D E
+-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
|9| |0| |8| |0| |8| |1| |8| |0| |8| |1|
+-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
| |*| | | |*| | | |*| | | |*| | | |*| |
+-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
|0| | | |0| | | |1| | | |1| | | |0| | |
+-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
]
Here is a general formula for solving the puzzle: (n² + 3n + 2)(n² +
3n + 3)/6. Whatever may be the stipulated number of residents along
each of the sides (which number is represented by n), the total number
of different arrangements may be thus ascertained. In our particular
case the number of residents was nine. Therefore (81 + 27 + 2) × (81 +
27 + 3) and the product, divided by 6, gives 2,035. If the number of
residents had been 0, 1, 2, 3, 4, 5, 6, 7, or 8, the total
arrangements would be 1, 7, 26, 70, 155, 301, 532, 876, or 1,365
respectively.