Recently I had the pleasure of attending a conference dedicated to the History of Recreational Mathematics where David Singmaster gave a wonderful talk on the problem of the 17 camels (and a related problem on 13 camels). The ideas below all came from his talk.
In a common version of the 17 camels problem a Sheikh dies, leaving his 17 camels to his 3 sons. The Sheikh decrees that his oldest son shall inherit one in two camels, his middle son shall inherit one in three camels and his youngest son shall inherit one in nine camels. The three don’t know what to do so they ask a wise man who advises them. What does he advise?
The wise man loans the sons his camel, thus making 18 camels to divvy up. The oldest son takes 9 camels, the second son takes 6 camels, the youngest son takes 2 camels and the wise man rides off on his original camel.
I had fun thinking up a similar version of the puzzle this morning:
After the sad death of farmer John a lawyer visited his four children to divide up his estate. John left 1/2 his estate to his firstborn, Alastair, 1/3 of his estate to his second child, Bertha, 1/10 of his estate to his third child, Colin, and a meagre 1/18 of his estate to the youngest child, Dorothy.
The lawyer announced that John was not the most successful farmer in the world (or even in West Dorset) and the only item to be divided up was a herd of 89 cattle. After a few moments of silent contemplation all four children leapt up and began arguing about how many cows they should each get and why their father had devised such a stupid way of dividing up his earthly belongings.
Bertha was mid-flow, bellowing words that I will not repeat here at Alastair when a wise old cow from a neighbouring farm entered the unhappy scene. She stepped forward and uttered,
“Stop this unnecessary beef and let me be of assistance”
“How, how?” exclaimed the children
“youuuuuuu cud add me to your herd and divide your father’s estate again”
After wondering quite how wise this wise old cow actually was the children agreed to add the cow to their herd. The lawyer smiled and bequeathed 45 cows to Alastair, 30 cows to Bertha, 9 cows to Colin and 5 cows to Dorothy, just as their father had requested.
The children were astonished to see the wise old cow hadn’t been assigned to any of them. The lawyer laughed and walked off with the wise old cow and he later made millions touring the world with his amazing talking cow act.
The classic story about a father how distributes his 17 camels as 1/2, 1/3, 1/9 works because
Just for fun we might ask, suppose there are n heirs, how can we set a version of the camels? We need to find natural numbers
This gives you a handle on how to construct amusing examples like the farmer John example above. The number of possible solutions grows rather quickly as you can see here.