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Odd Differences

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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Proofs with Pictures

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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For What?

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

Cosines Rule

Age 14 to 16 Challenge Level:

Why do this problem?
The first part is an easy application of the cosine rule. The second part requires some algebraic skills and gives practice in proving a formula.

Possible approach
Use as practice. The problem does not require learners to decide on the method to use.

Key questions
Look at the formula to be proved. Write down the cosine formula for the triangles in the diagram using the same notation. Can you see how to manipulate the formulas you have established to give the formula to be proved?