The number of ways in which the four pigs may be
placed in the
sties in accordance with the conditions is seventeen, including
example that I gave, not counting the reversals and reflections of
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
thathe was in
error, and, doubtless, had mistaken reversals for
Here are the seventeen answers.
The figures indicate the rows,
positions show the columns.
Thus, 104603 means that we place a pig in
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
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:
It will be found that forms N and Q are
the centre, and therefore give only two arrangements each by reversal
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
but there is only one way of doing it (if we do not count reversals as
different), as follows: 105030.