
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 twentyone.
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 twentyone 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 seventytwo other
numbers between these that the cat might employ with equal success.
