Alex's Adventures in Numberland - Alex Bellos [89]
As I was going to St Ives,
I met a man with seven wives,
Every wife had seven sacks,
Every sack had seven cats,
Every cat had seven kits.
Kits, cats, sacks, wives,
How many were going to St Ives?
The verse is the most famous trick question in English literature since, presumably, the man and his phalanx of females and confined felines were coming from St Ives. Irrespective of the direction of travel, however, the total number of kits, cats, sacks and wives is 7 + 72 + 73 + 74, which is 2800.
Another, less well-known, appearance of the riddle is as a problem in Leonardo Fibonacci’s Liber Abaci, from the thirteenth century. This version involved seven women on their way to Rome with increasing numbers of mules, sacks, loaves, knives and sheaths. The extra 76 brings the series to 137,256.
What is the allure of rising powers of seven that they have appeared in such different ages and contexts? Each case demonstrates the turbo-charged acceleration of geometric progressions. The rhyme is a poetic way of showing how quickly small numbers can lead to big ones. On first hearing, you think there might be a fair amount kits, cats, sacks and wives – but not almost 3000 of them! Likewise, the playful problems set in the Rhind Papyrus and the Liber Abaci express the same mathematical insight. And the number 7, though it would seem like it should have some special quality to make it so common across these problems, is rather irrelevant. When you multiply any number by itself a few times, the sum quickly reaches a counter-intuitively high amount.
Even when multiplying the lowest number possible, 2, by itself, the sum swirls to the heavens at a dizzying pace. Place one grain of wheat on the corner square of a chessboard. Place two grains on the adjacent square, and then start filling up the rest of the board by doubling the grains of wheat per square. How much wheat would you need to fill the final square? A few truckloads, or a container, maybe? There are 64 squares on a chessboard, so we have doubled up 63 times, meaning that the number is 2 multiplied by itself 63 times, or 263. In grains, this number is about 100 times more than the world’s current annual wheat production. Or, to consider it another way, if you started counting a grain of wheat per second at the very moment of the Big Bang 13 billion or so years ago, then you would not even have counted up to a tenth of 263 by now.
Mathematical riddles, rhymes and games are now collectively known as recreational maths. It is a wide-ranging and vibrant field, an essential feature of which is that the topics are accessible to the dedicated layperson, even though they might touch on impossibly complicated theory. Or they might not even involve theory at all, but rather merely kindle an appreciation of the wonder of numbers – such as the thrill of collecting pictures of licence plates.
A landmark event in the history of recreational maths is said to have taken place by the banks of the Yellow River in China around 2000 bc. According to legend, Emperor Yu saw a turtle creep out of the water. It was a divine turtle, with black and white dots on its underbelly. The dots denoted the first nine numbers and formed a grid on the turtle’s belly that (if the dots were instead written as Arabic numerals) looked like A:
A square like this one, that contains all the consecutive numbers starting from 1 and arranged so that all the rows, columns and corner-to-corner diagonals add up to the same total, is known as a magic square. The Chinese called this square the lo shu. (Its rows, columns and diagonals all add up to 15.) The Chinese believed that the lo shu symbolized the inner harmonies of the universe and used it for divination and worship. For example, if you start at 1 and draw a line between the numbers of the square in order, you map out the pattern that can be seen in B and the figure opposite,