by Burkard Polster and Marty Ross
The Age, 25 May 2009
If there are young children in your family then you are probably familiar with the word lists they bring home from school. The words on these lists are intended to be memorised, which can be pretty boring, especially for any adult who is assisting. Moreover, simply pointing at the words can result in the kids memorising the words by position on the list, but not otherwise.
Of course, gaining a vocabulary is extremely important. So here we present our mathemagical revamping of this valuable exercise.
Instead of making a standard list, arrange the words as the petals of a flower, as shown above. If you print this flower on a piece of cardboard, you can then throw it in the air with a spin. When you catch it, your thumb covers one of the words. You uncover the word and your little word wizard has to tell you what it is. Spinning the flower over and over will bring up the words in a random order.
Of course the spinning method will involve some repetition, and it may take a while for all the words to appear. Our favourite solution to this problem? Cheat: just surreptitiously slide your thumb to get that last word or two.
However, there is a nice alternative to spinning and cheating. Choose any starting word, and then have the child choose their favorite number, different from the number of petals. From your starting word, count the petals clockwise until this favorite number is reached, giving you your second word. From the second word, again count the favourite number around the petals to obtain your third word, and so on.
Notice that our pictured flower has 13 petals. (You will also notice that the children of maths masters practise a somewhat special vocabulary). Why 13? And no, for once the answer has nothing to do with Fibonacci numbers. The point is that 13 is a prime number.
Whatever the number of petals, eventually you will get back to the word you started with. Will you encounter all the words of the flower along the way? Yes, definitely, as long as you design your flower to have a prime number of petals. Moreover, in this case each petal will be visited exactly once.
Alas, most school lists do not consist of a prime number of words. What then? Well, you’re still safe as long as the list number and the chosen favourite number have no common factors.
For example, if your word flower has 15 petals then you still have the safe choices of 1, 2, 4, 7, 8, 11, 13, and 14. And if a different number is chosen instead? Then not all words will be reached. But the pattern of hit and miss is very interesting. Try it, say with 6, and see: mark a word each time it is chosen.
OK, that section was rated G. Now, something for the grown ups. Imagine that you have a flower with a 100-digit number of petals, and that your favourite number also has 100 digits. How do you decide whether these two numbers have a common factor? Finding the factors of large numbers is notoriously difficult, a critical assumption underlying the encryption of credit cards.
However, though we probably cannot factorise our two huge numbers, it is very easy to figure out whether they have a common factor. All we require is an ancient treasure of school arithmetic, the Euclidean algorithm.
By division, or even just repeated subtraction, we can calculate the remainder left over when the smaller of our numbers is divided into the larger. Repeating this process, the Euclidean algorithm allows us easily to find the greatest common factor of our two numbers.
This was already known to the ancient Greeks, even before Euclid. Now, 2500 years later, it is still one of the most useful, and one of the most simple and elegant algorithms in mathematics.
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