Reasoning, convincing and proving

  • Birthday Party
    problem

    Birthday party

    Age
    11 to 14
    Challenge level
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    The 30 students in a class have 25 different birthdays between them. What is the largest number that can share any birthday?
  • Wallpaper
    problem

    Wallpaper

    Age
    5 to 7
    Challenge level
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    These pieces of wallpaper need to be ordered from smallest to largest. Can you find a way to do it?

  • Sizing them Up
    problem

    Sizing them up

    Age
    5 to 7
    Challenge level
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    Can you put these shapes in order of size? Start with the smallest.

  • Trapezium Four
    problem

    Trapezium four

    Age
    14 to 16
    Challenge level
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    The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

  • Domino Sorting
    problem

    Domino sorting

    Age
    5 to 7
    Challenge level
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    Try grouping the dominoes in the ways described. Are there any left over each time? Can you explain why?

  • Integral Inequality
    problem

    Integral inequality

    Age
    16 to 18
    Challenge level
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    An inequality involving integrals of squares of functions.
  • Integral Sandwich
    problem

    Integral sandwich

    Age
    16 to 18
    Challenge level
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    Generalise this inequality involving integrals.
  • Product Sudoku
    problem

    Product Sudoku

    Age
    11 to 16
    Challenge level
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    The clues for this Sudoku are the product of the numbers in adjacent squares.

  • Litov's Mean Value Theorem
    problem

    Litov's mean value theorem

    Age
    11 to 14
    Challenge level
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    Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

  • Squares in rectangles
    problem

    Squares in rectangles

    Age
    11 to 14
    Challenge level
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    A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?