Expanding and factorising quadratics

  • Iff
    problem
    Favourite

    Iff

    Age
    14 to 18
    Challenge level

    Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

  • Always Perfect
    problem
    Favourite

    Always Perfect

    Age
    14 to 18
    Challenge level

    Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

  • Parabella
    problem
    Favourite

    Parabella

    Age
    16 to 18
    Challenge level

    This is a beautiful result involving a parabola and parallels.

  • Two Cubes
    problem

    Two Cubes

    Age
    14 to 16
    Challenge level
    Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]
  • Novemberish
    problem

    Novemberish

    Age
    14 to 16
    Challenge level
    a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.
  • Multiplication Magic
    problem

    Multiplication Magic

    Age
    14 to 16
    Challenge level

    Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). The question asks you to explain the trick.

  • Never Prime
    problem

    Never Prime

    Age
    14 to 16
    Challenge level

    If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

  • Composite Notions
    problem

    Composite Notions

    Age
    14 to 16
    Challenge level

    A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

  • Small tomato seedlings in pink pots.
    problem

    Leftovers

    Age
    14 to 16
    Challenge level

    Weekly Problem 26 - 2008
    If $n$ is a positive integer, how many different values for the remainder are obtained when $n^2$ is divided by $n+4$?

  • Geometric Parabola
    problem

    Geometric Parabola

    Age
    14 to 16
    Challenge level

    Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

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