exertion into his daily climbing down
as he did in his climbing up, and sleeping and slipping at night as
before."

This is the true version of the puzzle, and my readers will perhaps be
interested in working out the exact number of days. Of course, in a
puzzle of this kind the day is always supposed to be equally divided into
twelve hours' daytime and twelve hours' night.

107.--_The Four Princes._

The dominions of a certain Eastern monarch formed a perfectly square
tract of country. It happened that the king one day discovered that his
four sons were not only plotting against each other, but were in secret
rebellion against himself. After consulting with his advisers he decided
not to exile the princes, but to confine them to the four corners of the
country, where each should be given a triangular territory of equal area,
beyond the boundaries of which they would pass at the cost of their
lives. Now, the royal surveyor found himself confronted by great natural
difficulties, owing to the wild character of the country. The result was
that while each was given exactly the same area, the four triangular
districts were all of different shapes, somewhat in the manner shown in
the illustration. The puzzle is to give the three measurements for each
of the four districts in the smallest possible numbers--all whole
furlongs. In other words, it is required to find (in the smallest
possible numbers) four rational right-angled triangles of equal area.

[Illustration]

108.--_Plato and the Nines._

Both in ancient and in modern times the number nine has been considered
to possess peculiarly mystic qualities. We know, for instance, that there
were nine Muses, nine rivers of Hades, and that Vulcan was nine days
falling down from heaven. Then it has been confidently held that nine
tailors make a man; while we know that there are nine planets, nine days'
wonders, and that a cat has nine lives--and sometimes nine tails.

Most people are acquainted with some of the curious properties of the
number nine in ordinary arithmetic. For example, write down a number
containing as many figures as you like, add these figures together, and
deduct the sum from the first number. Now, the sum of the figures in this
new number will always be a multiple of nine.

There was once a worthy man at Athens who was not only a cranky
arithmetician, but also a mystic. He was deeply convinced of the magic
properties of the number nine, and