Mathematical_Recreations-Kraitchik-2e

28 Mathematical Recreations
24. Find a cube which is the sum of two cubes.
26. Express 10 as the sum of two numbers in such a way
that if each of them is divided by the other and the reSUlting
quotients are added, the final sum is equal to one of the two
original numbers.
26. Find three terms in continuous proportion whose sum
is a square.
27. Find a square which remains a square when it is increased
by 2, and when it is decreased by 2 plus its square
root.
The author (Beha Eddin Mohammed ben al Hosain al
Aamouli, 1547-1622) who collected these Arab problems remarks
that the answers to them are not known to the learned
persons of his time. In fact, some are impossible. For instance,
the impossibility of No. 24 was proved by Euler. It
is one case of the famous Fermat theorem: The equation
an + bn = c n has no solutions in integers if n is an integer
greater than 2. Others were solved by Euler, while still
others lead to equations of higher degree than the first which
have no positive rational solutions.
28. THE PROBLEM OF THE PANDECTS. A hungry hunter
came upon two shepherds, one of whom had 3 small loaves
of bread, and the other 5, all of the same size. The loaves
were divided equally among the three, and the hunter paid
8 cents for his share. How should the shepherds divide the
money? Answer: 1 and 7.
There are many variants of this problem. The following
(taken from an assortment in Unterrichtsblatter jur Mathematik
und N aturwissenschajten, xi, pp. 81-85), is probably
the version which gave the problem its name:
29. For their common meal Caius provided 7 dishes and