A Will of Diamonds

A man left $114$ identical diamonds, with instructions that they go to his four sons: one third to the eldest, one quarter to the second, one fifth to the third and one sixth to the youngest. No diamond was to be cut. But a third of $114$ is $38$, a quarter is not a whole number at all, and the executor was stuck. Can the will be carried out?

Solution

The fractions in the will do not add up to one: $$\frac13 + \frac14 + \frac15 + \frac16 = \frac{20 + 15 + 12 + 10}{60} = \frac{57}{60} = \frac{19}{20}.$$ So the father never disposed of his whole estate, only nineteen twentieths of it, and that is what makes $114$ awkward to divide. The cure is the old trick of the seventeen camels. Borrow six diamonds from a friend, making $120$. Now the shares come out whole: $$\frac{120}{3} = 40, \quad \frac{120}{4} = 30, \quad \frac{120}{5} = 24, \quad \frac{120}{6} = 20,$$ and these total $40 + 30 + 24 + 20 = 114$. Give the sons $40$, $30$, $24$ and $20$ diamonds, hand the six borrowed stones back, and every son has received rather more than the letter of the will would have given him, with not a single diamond cut. The borrowed six were never needed except to make the arithmetic whole, which is exactly why $120$, the lowest common multiple of the denominators, is the number to reach for.