THOSE FIFTEEN SHEEP.
(
Combination and Group Problems)
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.
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 in
the cyclopaedia. What was it? Can you place those fifteen sheep?
Answer:
If we read the exact words of the writer in the cyclopaedia, we find that
we are not told that the pens were all necessarily empty! In fact, if
the reader will refer back to the illustration, he will see that one
sheep is already in one of the pens. It was just at this point that the
wily farmer said to me, "_Now_ I'm going to start placing the fifteen
sheep." He thereupon proceeded to drive three from his flock into the
already occupied pen, and then placed four sheep in each of the other
three pens. "There," says he, "you have seen me place fifteen sheep in
four pens so that there shall be the same number of sheep in every pen."
I was, of course, forced to admit that he was perfectly correct,
according to the exact wording of the question.
