CATCHING THE MICE.
(
Moving Counter Problem)
"Play fair!" said the mice. "You know the rules of the game."
"Yes, I know the rules," said the cat. "I've got to go round and round
the circle, in the direction that you are looking, and eat every
thirteenth mouse, but I must keep the white mouse for a tit-bit at the
finish. Thirteen is an unlucky number, but I will do my best to oblige
you."
"Hurry up, then!" shouted the mice.
"Give a fellow time to think," said the cat. "I don't know which of you
to start at. I must figure it out."
While the cat was working out the puzzle he fell asleep, and, the spell
being thus broken, the mice returned home in safety. At which mouse
should the cat have started the count in order that the white mouse
should be the last eaten?
When the reader has solved that little puzzle, here is a second one for
him. What is the smallest number that the cat can count round and round
the circle, if he must start at the white mouse (calling that "one" in
the count) and still eat the white mouse last of all?
And as a third puzzle try to discover what is the smallest number that
the cat can count round and round if she must start at the white mouse
(calling that "one") and make the white mouse the third eaten.
Answer:
In order that the cat should eat every thirteenth mouse, and the white
mouse last of all, it is necessary that the count should begin at the
seventh mouse (calling the white one the first)--that is, at the one
nearest the tip of the cat's tail. In this case it is not at all
necessary to try starting at all the mice in turn until you come to the
right one, for you can just start anywhere and note how far distant the
last one eaten is from the starting point. You will find it to be the
eighth, and therefore must start at the eighth, counting backwards from
the white mouse. This is the one I have indicated.
In the case of the second puzzle, where you have to find the smallest
number with which the cat may start at the white mouse and eat this one
last of all, unless you have mastered the general solution of the
problem, which is very difficult, there is no better course open to you
than to try every number in succession until you come to one that works
correctly. The smallest number is twenty-one. If you have to proceed by
trial, you will shorten your labour a great deal by only counting out
the remainders when the number is divided successively by 13, 12, 11,
10, etc. Thus, in the case of 21, we have the remainders 8, 9, 10, 1, 3,
5, 7, 3, 1, 1, 3, 1, 1. Note that I do not give the remainders of 7, 3,
and 1 as nought, but as 7, 3, and 1. Now, count round each of these
numbers in turn, and you will find that the white mouse is killed last
of all. Of course, if we wanted simply any number, not the smallest, the
solution is very easy, for we merely take the least common multiple of
13, 12, 11, 10, etc. down to 2. This is 360360, and you will find that
the first count kills the thirteenth mouse, the next the twelfth, the
next the eleventh, and so on down to the first. But the most
arithmetically inclined cat could not be expected to take such a big
number when a small one like twenty-one would equally serve its purpose.
In the third case, the smallest number is 100. The number 1,000 would
also do, and there are just seventy-two other numbers between these that
the cat might employ with equal success.
