Reasoning, convincing and proving

  • Your number is...
    problem

    Your number is...

    Age
    7 to 14
    Challenge level
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    Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?

  • Squirty
    problem

    Squirty

    Age
    14 to 16
    Challenge level
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    Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

  • Break it up!
    problem

    Break it up!

    Age
    5 to 11
    Challenge level
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    In how many different ways can you break up a stick of seven interlocking cubes? Now try with a stick of eight cubes and a stick of six cubes. What do you notice?

  • More Number Pyramids
    problem

    More number pyramids

    Age
    11 to 14
    Challenge level
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    When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

  • Odd Squares
    problem

    Odd squares

    Age
    7 to 11
    Challenge level
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    Think of a number, square it and subtract your starting number. Is the number you're left with odd or even? How do the images help to explain this?

  • Generally Geometric
    problem

    Generally geometric

    Age
    16 to 18
    Challenge level
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    Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
  • Circle Box
    problem

    Circle box

    Age
    14 to 16
    Challenge level
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    It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
  • An Unusual Shape
    problem

    An unusual shape

    Age
    11 to 14
    Challenge level
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    Can you maximise the area available to a grazing goat?

  • Inscribed in a Circle
    problem

    Inscribed in a circle

    Age
    14 to 16
    Challenge level
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    The area of a square inscribed in a circle with a unit radius is 2. What is the area of these other regular polygons inscribed in a circle with a unit radius?

  • Special Numbers
    problem

    Special numbers

    Age
    11 to 14
    Challenge level
    filled star empty star empty star

    My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?