Answer :
The number of ways
in which the four pigs may be
placed in the
thirtysix
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 twentyone
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
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
semisymmetrical with
regard to
the centre, and therefore give only two arrangements each by reversal
and
reflection; that form H is quartersymmetrical, 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.
