Identities

problem
Favourite

Binomial

Age

16 to 18

Challenge level

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
problem
Favourite

Cubes Within Cubes Revisited

Age

11 to 14

Challenge level

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
problem
Favourite

Partitioning Revisited

Age

11 to 14

Challenge level

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
problem

Powerful Factors

Age

16 to 18

Challenge level

Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.
problem

Particularly General

Age

16 to 18

Challenge level

By proving these particular identities, prove the existence of general cases.
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Proofs With Pictures

Some diagrammatic 'proofs' of algebraic identities and inequalities.