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 was perpetually strolling out to the groves of Academia to bother poor old Plato with his nonsensical ideas about what he called his "lucky number." But Plato devised a way of getting rid of him. When the seer one day proposed to inflict on him a lengthy disquisition on his favourite topic, the philosopher cut him short with the remark, "Look here, old chappie" (that is the nearest translation of the original Greek term of familiarity): "when you can bring me the solution of this little mystery of the three nines I shall be happy to listen to your treatise, and, in fact, record it on my phonograph for the benefit of posterity." [Illustration] Plato then showed, in the manner depicted in our illustration, that three nines may be arranged so as to represent the number eleven, by putting them into the form of a fraction. The puzzle he then propounded was so to arrange the three nines that they will represent the number twenty.