快猫短视频

#68 Diamonds

鈥淥nce upon a time,鈥� began Ivan the storyteller, with children at his feet, 鈥渢here 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: 鈥業 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.'鈥� Ivan reached into his pocket and pulled out a small wooden casket. 鈥淎nd this is the box!鈥� 鈥淗ow many diamonds are there?鈥� screamed the children. 鈥淚f you can tell me, I will give you the box,鈥� said Ivan. 鈥淏ut you haven鈥檛 told us how many palaces鈥︹�� they cried. Ivan winked.

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

Answer next week

#67 My prime

Solution

鈥淢y prime鈥� 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鈥檚 larger digit 鈥渁鈥� and the smaller 鈥渂鈥�. The difference between their two squares is a虏 鈥� b虏, which is the same as (a + b) x (a 鈥� 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 鈥� b, the difference between the digits, is equal to 1.