The Four Porkers
(
MISCELLANEOUS PUZZLES)
The four pigs are so placed, each in a separate sty, that although every one of the thirty-six sties is in a straight line (either horizontally, vertically, or diagonally), with at least one of the pigs, yet no pig is in line with another. In how many different ways may the four pigs be placed to fulfil these conditions? If you turn this page round you get three more arrangements, and if you turn it round in front of a mirror you get four more. These are not to be counted as different arrangements.
Answer:
The number of ways in which the four pigs may be placed in the thirty-six sties in accordance with the conditions is seventeen, including the example that I gave, not counting the reversals and reflections of these arrangements as different. Jaenisch, in his Analyse Mathématique au jeu des Échecs (1862), quotes the statement that there are just twenty-one solutions to the little problem on which this puzzle is based. As I had myself only recorded seventeen, I examined the matter again, and found that he was in error, and, doubtless, had mistaken reversals for different arrangements.
Here are the seventeen answers. The figures indicate the rows, and their positions show the columns. Thus, 104603 means that we place a pig in the first row of the first column, in no row of the second column, in the fourth row of the third column, in the sixth row of the fourth column, in no row of the fifth column, and in the third row of the sixth column. The arrangement E is that which I gave in diagram form:—
A. |
104603 |
B. |
136002 |
C. |
140502 |
D. |
140520 |
E. |
160025 |
F. |
160304 |
G. |
201405 |
H. |
201605 |
I. |
205104 |
J. |
206104 |
K. |
241005 |
L. |
250014 |
M. |
250630 |
N. |
260015 |
O. |
261005 |
P. |
261040 |
Q. |
306104 |
It will be found that forms N and Q are semi-symmetrical with regard to the centre, and therefore give only two arrangements each by reversal and reflection; that form H is quarter-symmetrical, and gives only four arrangements; while all the fourteen others yield by reversal and reflection eight arrangements each. Therefore the pigs may be placed in (2 × 2) + (4 × 1) + (8 × 14) = 120 different ways by reversing and reflecting all the seventeen forms.
Three pigs alone may be placed so that every sty is in line with a pig, provided that the pigs are not forbidden to be in line with one another; but there is only one way of doing it (if we do not count reversals as different), as follows: 105030.
Informational
