A story of apples

"There once lived in Damascus an enterprising peasant who had three daughters. One day, the peasant told a qadi, a judge, that his daughters were not only very intelligent but blessed with rare skills of the imagination. The qadi, a jealous and stingy man, was annoyed at hearing a peasant speak so praisingly of his daughters' talents. [...] The qadi had the three girls brought before him. Then he said to them, 'Here are 90 apples for you to sell in the market. Fatima, the oldest, you will take 50; Cunda, you will take 30; and Shia, the youngest, you will take 10. If Fatima sells the apples at a price of 7 to the dinar, you other two will have to sell yours at the same price. And if Fatima sells her apples for 3 dinars each, you two will have to do the same. But, no matter what you do, each of you must end up with the same amount of money from your different numbers of apples.'

" 'But can I not give away some of the apples that I have?' Fatima asked.

" 'Under no circumstances,' said the wretched qadi. 'These are the terms: Fatima must sell 50 apples. Cunda must sell 30 apples. And Shia must sell the 10 apples that remain. And all of you must sell your apples at the same price, and all of you must earn exactly the same profit in the end.' "

Such was the problem posed to our class by The Man Who Counted this morning. The kids ultimately came up with three solutions, and while none were the same as that presented by Beremiz, in the book, each was an example of creative and intelligent problem solving:

— The girls could sell their apples in unequal lots of equal price. So, Fatima could sell all her apples in a single lot of 50 for any price she chose; 15 dinars, for example. Then Cunda would sell her 30 apples in a single lot for 15 dinars, and Shia would do the same with her 10.

— The girls could combine their apples into a single lot (e.g. 90 apples sold en masse for 30 dinars), then split the profits evenly.

— Owning and selling are not the same thing. The girls could, on leaving the presence of the qadi, divide the apples evenly among themselves, deciding that each sister owned 30 apples. Upon arriving at the market, Shia could then hand 20 of her apples to Fatima to sell, on the understanding that the profits of the same would be handed back to Shia, the owner of the apples. Then, Shia would sell 10, Fatima would sell 50, and Cunda would sell 30 apples, and each receive the same profit.

What ingenious solutions!

For the curious, here is the solution presented by Beremiz:

"Fatima starts selling her apples at a price of 7 apples for 1 dinar. She sells 49 apples at this price, but keeps back 1.

"Cunda sells 28 of her apples at this price, but keeps back 2.

"Shia sells 7 of her apples at this price, but keeps back 3.

"Then Fatima sells her 1 remaining apple for 3 dinars. In accordance with the rules of the qadi, Cunda then sells her 2 remaining apples for 3 dinars each. And Shia then sells her 3 remaining apples for 3 dinars each. [...] Therefore, each made a profit of 10 dinars, and thus the problem set by the envious qadi of Damascus was solved."