Identities

  • Cubes within Cubes revisited
    problem
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    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?
  • Partitioning revisited
    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

  • Binomial
    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

  • Powerful Factors
    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.

  • Particularly general
    problem

    Particularly General

    Age
    16 to 18
    Challenge level

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

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