Neurobics

The Perimeter Problem

Russell Dear

A student who had been given the task of finding areas and perimeters of rectangles told me he had found one with its perimeter numerically equal to its area. It happened to be the four-by-four square. In commending him I mistakenly said that he'd found the only rectangle with sides whole numbers that had that property. He very quickly proved me wrong. Can you find the other solution? There is only one other.

On the subject of perimeters you might find this problem a little more challenging.

A unit square has a perimeter of four units. When two such squares are placed, non-overlapping, in a plane, the minimum perimeter of the shape they form is six units. For three squares the minimum perimeter possible is eight units. Perimeter = 6 units
Perimeters = 8 units

What is the minimum perimeter when (i) four unit squares are placed, non-overlapping, in a plane? (ii) How about five squares? (iii) Six squares? (iv) 100 squares?

Can you generalise for shapes formed from n non-overlapping unit squares in a plane?

Answers

Russell Dear is a Mathematician living in Invercargill