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#68 Diamonds

“Once upon a time,” began Ivan the storyteller, with children at his feet, “there lived a queen called Factoria who had six daughters and many palaces. In each palace, she kept as many crystal vases as she had palaces, and in each vase were as many diamonds as there were vases in that palace. Then one day the Queen died, leaving a will: ‘I leave one vase of diamonds to my loyal servant Fidelio. The rest of the diamonds I will share equally between my daughters. Any remaining diamonds, Fidelio will put in this box.'” Ivan reached into his pocket and pulled out a small wooden casket. “And this is the box!” “How many diamonds are there?” screamed the children. “If you can tell me, I will give you the box,” said Ivan. “But you haven’t told us how many palaces…” they cried. Ivan winked.

How many diamonds are in the box? How can you be certain?

Answer next week

#67 My prime

Solution

“My prime” could be 23, 43, 67 or 89, but whichever it is, the two digits must differ by 1.

You could discover this by trial and error, but there is a short cut. Let us call the number’s larger digit “a” and the smaller “b”. The difference between their two squares is a² – b², which is the same as (a + b) x (a – b). However, we were told this difference is a prime number, so the only two factors of ab are itself and 1. Therefore a + b must be the prime itself, while a – b, the difference between the digits, is equal to 1.