# Purely fizzical;Mathematics

Fizz-buzz is an old arithmetical game. You arrange your class in some sort of cyclic order, and they count in turn from 1 onwards, except that every time they meet a multiple of 3 they say "fizz". The next stage is to include, say, multiples of 4, for which they say "buzz". They then notice that when they get to 12 they have to say "fizz-buzz".

I have played this many times with eight-year-olds, saying "bing" for multiples of 2, "bang" for multiples of 3, and eventually adding "bong" for multiples of 5, and you have to keep at it long enough to get at least to the first "bing-bang-bong". But a group of 10 and 11-year-olds I visited recently had been experimenting with a more complicated version of the game. "Fizz" was for multiples of 5 and "buzz" was for multiples of 7, but they also added a "fizz" or a "buzz" if the respective digits occurred in the number. So, they explained to me, 35 would be "fizz-fizz-buzz" (a fizz and a buzz for the multiples and an extra fizz for the digit 5), and 25 would be "fizz-fizz-fizz" (two fizzes for the two divisors of 5 and one for the digit).

An effective problem-creating strategy is that of turning things back to front. It is fairly straightforward to start with numbers and transform them into the fizz-buzz code according to divisors and digits, but it requires far more thought if you reverse the procedure and ask, as I did, for a number that was a "fizz-fizz-buzz-buzz".

Andrew soon came up with 775, and explained to the sceptics it had two 7s and a 5, and was also a multiple of 5.

I could have asked for other numbers that satisfied this, but instead I asked for the lowest number that was a "fizz-buzz".

Several now thought of 57, neatly including both digits without creating a multiple of either. A suggestion of 75 was dismissed because it was also a multiple of 5, so it was a "fizz-fizz-buzz". I asked if there was a lower number than 57, and in the face of general agreement that there was not, I had to insist there was.

Some suggested 35, in spite of the previous discussion about this, and had it refuted by others.

Andrew insisted it had to be a multiple of 7 with a 5 in it, and the only one was 35. I am not sure what he was thinking of, but his authority seemed to impress the others. It took some time for Ken to produce 56.

Another problem-creating strategy is to ask for all the possibilities in a situation, and this reverse problem lends itself to such a strategy.

Suppose you ask for all the numbers represented by "fizz". The first is obviously 5. To get higher numbers you need a multiple of 5 with no 5-digits, or a number with a 5-digit which is not a multiple of 5. The first category cannot include odd multiples of 5, because each has a 5 at the end. So we need any even multiple of 5 - that is, a multiple of 10 - that does not contain a 5. But we have to be careful, because 100, for instance, is a "fizz-fizz". There is no formula as such, but we can either describe the set of numbers this way (all multiples of 10 that do not contain a 5 and are not multiples of 100) or we can list those, say, that are under 1,000.

The second category is slightly easier, because we can first consider any number containing just one 5-digit, as long as that digit is not at the end. But we have to avoid multiples of 25, because they would be "fizz-fizzes". How can we do that?

Finding "buzz-buzz" numbers is a different kind of problem, because multiples of 7 do not behave like multiples of 5 as far as the digits are concerned.

In the same way, multiples of other numbers will have different characteristics. Further explorations can involve pairs of numbers with a common divisor, such as 4 and 6; or pairs, one of which divides the other, such as 3 and 9; or three multiples or more.

This sort of work demands a variety of problem-solving skills. It also encourages a deeper awareness of the relationships between multiples and divisors, and develops some understanding of the characteristics of different multiples when they are written in our place value system, and therefore their relationships to powers of 10.

* David Fielker is a freelance lecturer and writer and teaches part time at the American Community School, Egham, Surrey. Problem-solving strategies including the ones in this article are explored in his book, 'Extending Mathematical Ability', published by Hodder amp; Stoughton, pound;11.99