Diophantus II

We’re not actually sure how old this problem is. But it is sure that it is older than the one from a Lady’s Diary (See Algebra, Level 5, A Lady’s Age.) Now that may appear old to you but similar problems can be found in Indian writings. In one book, Lilivati , which was written around 1150 AD, the author, Bhaskara writes problems for his daughter to solve. There is rather an elegant question there about a peacock and a snake that involves Pythagoras’ Theorem.

Surprisingly the Kamasutra, which is probably famous for other things, contains a number of mathematical problems. I’m sure that you’ll read it with a new insight now. But problems in lay areas go back even further. In Diophantus I and II, we look at problems that were supposedly engraved on tombstones. We don’t know exactly when Diophantus actually lived but he was certainly around a couple of centuries or so before Christ.

We don’t know exactly when Diophantus actually lived but he was definitely around somewhere between 150 BC and 364 AD. (How’s that for a margin of error?) We also know that he lived in Alexandria which was the centre of ancient Greek civilization from 350 BC to 640 AD. Diophantus’ main claim to fame came from his thirteen-volume set of books called Arithmetica. Only six of them have survived but they tell us that he was interested in problems that had whole number solutions. Equations of this type are called Diophantine equations today in his honour.

If you want to look into a bit of mathematical history, we suggest that you look at the book A History of Mathematics by Carl Boyer. If you click his name you’ll be able to read a review of the book.

So there are many places in history where we can find mathematical problems in books that were read by non-mathematicians. Victorians seem to have enjoyed a good problem in the equivalent of their Woman’s Weekly. We guess that it was one way of whiling away the time, like doing crossword puzzles. Certainly recreational curiosities with a mathematical bent have had a market. Perhaps the most famous of the older ones of these are H.E. Dudeney’s Amusements in Mathematics, Nelson, 1919, W.W.R. Ball’s Mathematical Recreations and Essays, Macmillan, 1939. The French also got into the act with books by the well-respected mathematician, E. Lucas (the four volumes of his Récréations Mathématiques were published by Gautier-Villars, between 1883 and 1894), and others. Such books are still being written. Maybe the most famous of these are Martin Gardner’s many books (see, for example, Mathematical Puzzles and Diversions, More Mathematical Puzzles and Diversions, and Mathematical Carnival, published by Pelican Books) that came from the pages of the Scientific American.

Having said that, this problem is one of a set of three that involve similar techniques. The other problems in this trilogy at Algebra Level 6, are Lady’s Age and Diophantus I. All can be solved by guess and check or by being systematic in some way. However, the most efficient way to solve each one of them is by algebra. It may be worthwhile to allow your class to work on these problems using non-algebraic techniques first. Then they should be impressed by the power of algebra to solve them very efficiently.