Sometimes you know you are right. Often you can check with the back of the book, or the teacher. Mathematicians, and other people who use mathematics in their work, can't look up the back of the book. They might even be the teacher, rather than having one to ask! They have to decide just how confident they are of their answer.

These are problems to help you to consider your confidence levels and discuss how you feel about your solutions.

Theorem One: Sums of Squares

Theorem One:
27 is the smallest number which can be written as the sum of 3 squares of positive numbers in 2 different ways.
1. Can you write 27 as the sum of 3 squares in 2 different ways?
Do you agree with the theorem that 27 is the SMALLEST number which can be written as the sum of three squares? Why?
How confident are you?
What would make you more confident or are you absolutely sure?

Theorem Two: Sums of Cubes

Theorem Two:
There are only four numbers less than 100 which can be written as the sum of 3 different cubes - that is all cubes of different numbers.
2. Can you find four numbers less than 100 which can be written as the sum of three different cubes?
Do you agree with the theorem that these are the only four?
How confident are you? Why?

3. Which of the two problems looked easier? Which was easier? Why?


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I'm not going to ask you to check Theorem Three, you will be very pleased to know. But first a story from mathematical archives.  Srinivasa Ramanujan and G.H. Hardy are two men you will hear of many times if you are lucky. Ramanujan was a mathematical genius from India who came to Cambridge University in England to study with G.H. Hardy, who had recognised his genius. Not all went perfectly and Ramanujan died young, back in India, after much illness. When he was in hospital in England, Hardy visited him and commented that the number of the taxi cab he had just ridden in, Number 1729, was an "uninteresting number." Ramanujan disagreed. Did Hardy realise, he asked, that 1729 is the smallest integer which can be expressed as the sum of two cubes in two different ways? 

Theorem Three:
The smallest integer which can be expressed as the sum of cubes in two different ways is 1729.

Srinivasa Ramanujan, 1887 - 1920

Aren't you glad I didn't ask you to find that one?

Just in case you were wondering, 1729 is the sum of the cubes of 10 and 9, as well as being the sum of the cubes of 12 and 1. The sort of thing most people would think of when sick in hospital!