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.