NZSM Online

Get TurboNote+ desktop sticky notes

Interclue makes your browsing smarter, faster, more informative

SciTech Daily Review

Webcentre Ltd: Web solutions, Smart software, Quality graphics

Neurobics

Cutting Up Squares

By Russell Dear

Imagine a square of paper. Can you cut it up into smaller squares? It certainly doesn't sound difficult...

A square can be cut into:
Four squares... or six squares.

Nobody required that the squares had to be the same size.

Now, can we cut our square piece of paper into any and every number of squares? That means exactly, of course, with no paper left over.

In the jargon of dissection theory a square that has been cut into squares, not necessarily of the same size, is called a "squared square". A much more difficult problem is to cut a square into squares of different sizes. We then have what is called a "perfect square".

For a long time it was thought impossible to produce a perfect square until William Tutte and friends from Cambridge University did so in 1940. The smallest perfect square has a side of 175 units and can be cut into 24 unequal squares.

Another problem associated with squared squares is that of minimising the number of squares into which a square is cut. This type of problem is often called "Mrs Perkin's quilt" after an example of one named by Henry Dudeney in his book Amusements in Mathematics (1917).

Dudeney's problem is essentially this: Given a square 13 units by 13 units, what is the minimum number of squares into which it can be cut? All the squares are to be a whole number of units wide.

To get a feel for the problem look at squares with sides smaller than 13.

For squares of sides 2, 3, 4, 5 and 7 units the minimum number of squares in the dissection are 4, 6, 4, 8 and 9 as shown.

Consider the following questions:

1. Why was the square of side 6 units omitted from the above set of diagrams?

2. Why do only squares with sides of prime numbered length need to be considered?

3. What is the minimum number of squares into which a square of side 11 units can be dissected?

4. What is the solution of Dudeney's quilt problem?

A bit of playing around with paper and scissors may convince you that the answer to the original question about dividing a square into different numbers of smaller squares is that it is possible for just about every number of squares above five. How can you be absolutely sure? The following diagrams show that it is possible to cut a square into seven and eight squares

Six squares have also been shown. Can you see how seven squares have been obtained from four? Can you see now why it is possible to cut a square into any number of squares over five?

Answers

1. It is twice the size of the 3 by 3 square and therefore can be cut into four.
2. Non-prime squares are multiples of smaller squares.
3. 11
4. 11.

Russell Dear is a Mathematician living in Invercargill