13 = 579; 4821 - 83 = 4738; and so on. The numbers in this last
column give the required solutions. Thus, four husbands may be seated in
two ways, five husbands may be placed in thirteen ways, and six husbands
in eighty ways.

The following method, by Lucas, will show the remarkable way in which
chessboard analysis may be applied to the solution of a circular problem
of this kind. Divide a square into thirty-six cells, six by six, and
strike out all the cells in the long diagonal from the bottom left-hand
corner to the top right-hand corner, also the five cells in the diagonal
next above it and the cell in the bottom right-hand corner. The answer
for six couples will be the same as the number of ways in which you can
place six rooks (not using the cancelled cells) so that no rook shall
ever attack another rook. It will be found that the six rooks may be
placed in eighty different ways, which agrees with the above table.

262.--THOSE FIFTEEN SHEEP.

A certain cyclopaedia has the following curious problem, I am told:
"Place fifteen sheep in four pens so that there shall be the same number
of sheep in each pen." No answer whatever is vouchsafed, so I thought I
would investigate the matter. I saw that in dealing with apples or
bricks the thing would appear to be quite impossible, since four times
any number must be an even number, while fifteen is an odd number. I
thought, therefore, that there must be some quality peculiar to the
sheep that was not generally known. So I decided to interview some
farmers on the subject. The first one pointed out that if we put one pen
inside another, like the rings of a target, and placed all sheep in the
smallest pen, it would be all right. But I objected to this, because you
admittedly place all the sheep in one pen, not in four pens. The second
man said that if I placed four sheep in each of three pens and three
sheep in the last pen (that is fifteen sheep in all), and one of the
ewes in the last pen had a lamb during the night, there would be the
same number in each pen in the morning. This also failed to satisfy me.

[Illustration]

The third farmer said, "I've got four hurdle pens down in one of my
fields, and a small flock of wethers, so if you will just step down with
me I will show you how it is done." The illustration depicts my friend
as he is about to demonstrate the matter to me. His lucid explanation
was evidently that which was in the mind of the writer of the article