# Puzzlemaster

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The Independent Culture
I WAS saying that mathematics is not common sense. Those so-called popularisers that talk of it glibly as if it were obvious do it and us a disservice. There is the implication that if you do not understand things straightaway, you are a bit of a thicko.

In fact Nature has designed us to be thickos first and mathematicians second. Evolution has selected us more for our ability to survive than to philosophise; for our prejudices rather than the truth.

The effort needed to realign our prejudices with reality reflects the measure of this divergence. But if God has given us boz-eyes he has also given us in the objective sciences the mental means of correcting for them. In the sciences, predispositions such as common sense are subjected to sceptical enquiry. Proofs are demanded for the most "obvious" things. No wonder maths seems a little Zen. Indeed to many, the Zen-ness is its principal attraction.

This non-obviousness matters. It is important to realise not just why we study maths but also what to expect of it. I shudder now to think of all my contemporaries who were put off maths and science because they found it anti-intuitive and took this to mean they had a lack of talent for it.

Difficulties can be a sign of intelligence. And mistakes can be learning opportunities. It is how we recode our mistakes that marks our intelligence. It is the thrill of this adjustment that is the challenge and the fun. But it is no use tub-thumping and merely stating maths to be fun. You can only find that maths is fun by doing some. By active learning. Enough tub-thumping. This week's active learning:

Points to ponder

1 Which digit occurs most and which least in the numbers from 1 to 100 inclusive? Find formulae for the number of times a digit appears in the numbers from 1 to 10n inc. (10n means 1 followed by n zeros.)

2 An equilateral triangle can be divided into 4 equal equilateral triangles. Into what other numbers is such equal division possible?

Solutions to last week's puzzles

1 The joke I had in mind on the theme of "only just" was: A miser is suspiciously counting his change in a shop. "Is it right?" asks the shopkeeper. "Yes," replies the miser. "But only just."

In a similar vein, there's the bishop who, during a school visit, tests a little boy's mental arithmetic by asking what 3 X 7 is. Boy: 21. Bishop: Good... Boy: Good? It's bloody perfect!

2 The equilateral triangle in (a) has been divided into 11 equilateral triangles. Now design a procedure for dividing it instead into a) any even number (but not 2) b) any odd number (but not 3 or 5).

3 Instead of trying to divide the shape shown into four figures similar to the outline, try arranging four such tiles to make a similar shape. Note that three of the shapes have had to be flipped over (b).

comments to: indy@puzzlemaster.co.uk