Neurobics

Sums of Squares

Russell Dear

Some numbers can be expressed as the sum of one or two squares. For example; 9 = 3², 10 = 1² + 3² and 20 = 2² + 4².

Just under half of the whole numbers from one to a hundred can be expressed in this way. Here are four "search and find" problems associated with this property, although there are a number of short cuts you could use to make things easier.

(1) What is the length of the longest string of consecutive whole numbers less than 100 which cannot be expressed as the sum of one or two squares?

(2) How many whole numbers less than 100 cannot be expressed as the sum of three or less squares?

(3) What are the two smallest consecutive whole numbers that cannot be expressed as the sum of three or less squares?

(4) How many whole numbers less than one million cannot be expressed as the sum of four or less squares?

Russell Dear is a Mathematician living in Invercargill