
Answer :
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
lefthand 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 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
lefthand 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 lefthand 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 lefthand 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.
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.
