itions. If, again, it should be held that
the sisters might not, according to the wording, have kissed their
brother, although he kissed them, I reply that in that case there must
have been twelve girls, all of whom must have been his sisters. And the
reference to the ladies who might have worked exclusively of the sisters
shuts out the possibility of this.

106.--_The Adventurous Snail._

At the end of seventeen days the snail will have climbed 17 ft., and at
the end of its eighteenth day-time task it will be at the top. It
instantly begins slipping while sleeping, and will be 2 ft. down the
other side at the end of the eighteenth day of twenty-four hours. How
long will it take over the remaining 18 ft.? If it slips 2 ft. at night
it clearly overcomes the tendency to slip 2 ft. during the daytime, in
climbing up. In rowing up a river we have the stream against us, but in
coming down it is with us and helps us. If the snail can climb 3 ft. and
overcome the tendency to slip 2 ft. in twelve hours' ascent, it could
with the same exertion crawl 5 ft. a day on the level. Therefore, in
going down, the same exertion carries it 7 ft. in twelve hours--that is,
5 ft. by personal exertion and 2 ft. by slip. This, with the night slip,
gives it a descending progress of 9 ft. in the twenty-four hours. It can,
therefore, do the remaining 18 ft. in exactly two days, and the whole
journey, up and down, will take it exactly twenty days.

107.--_The Four Princes._

When Montucla, in his edition of Ozanam's _Recreations in Mathematics_,
declared that "No more than three right-angled triangles, equal to each
other, can be found in whole numbers, but we may find as many as we
choose in fractions," he curiously overlooked the obvious fact that if
you give all your sides a common denominator and then cancel that
denominator you have the required answer in integers!

Every reader should know that if we take any two numbers, _m_ and _n_,
then _m_^2 + _n_^2, _m_^2 - _n_^2, and _2mn_ will be the three sides of a
rational right-angled triangle. Here _m_ and _n_ are called generating
numbers. To form three such triangles of equal area, we use the following
simple formula, where _m_ is the greater number:--

_mn_ + _m_^2 + _n_^2 = _a_
_m_^2 - _n_^2 = _b_
2_mn_ + _n_^2 = _c_

Now, if we form three triangles from the following pairs of generators,
_a_ and _b_, _a_ and _c_, _a_ and _b_ + _c_, they will all be of equal
area. Th