Answer
:
The Monk might have
placed dogs in the kennels in
two thousand
nine
hundred and twentysix different ways, so that there should be ten dogs
on every side.
The number of dogs might vary from twenty to forty, and
as long as the Monk kept his animals within these limits the thing was
always possible.
The general
solution to this puzzle is difficult.
I find that
for n
dogs on every side of the square, the number of different ways is (n^{4}
+ 10n^{3}
+ 38n^{2}
+ 62n
+ 33) / 48, where n
is
odd, and ((n^{4}
+
10n^{3}
+ 38n^{2}
+ 68n)
/ 48) + 1, where n
is
even, if we count only
those arrangements that are fundamentally different.
But if we count
all
reversals and reflections as different, as the Monk himself did, then n
dogs (odd or even) may be placed in ((n^{4}
+ 6n^{3}
+ 14n^{2}
+ 15n)
/
6) + 1 ways.
In order that there
may be n
dogs on
every side, the
number must not be less than 2n
nor greater than 4n,
but it may be
any number within these limits.
