Explaining, convincing and proving

  • Angles inside
    problem
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    Angles inside

    Age
    11 to 14
    Challenge level
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    Draw some angles inside a rectangle. What do you notice? Can you prove it?

  • Calendar Capers
    problem

    Calendar capers

    Age
    11 to 14
    Challenge level
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    Choose any three by three square of dates on a calendar page...
  • Adding all nine
    problem

    Adding all nine

    Age
    11 to 14
    Challenge level
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    Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!
  • American Billions
    problem
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    American billions

    Age
    11 to 14
    Challenge level
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    Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

  • Blue and White
    problem
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    Blue and white

    Age
    11 to 14
    Challenge level
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    Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

  • Dicing with numbers
    problem

    Dicing with numbers

    Age
    11 to 14
    Challenge level
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    In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

  • The Genie in the Jar
    problem

    The genie in the jar

    Age
    11 to 14
    Challenge level
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    This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal spoons. Each day a spoonful was used to perfume the bath of a beautiful princess. For how many days did the whole jar last? The genie's master replied: Five hundred and ninety five days. What three numbers do the genie's words granid, ozvik and vaswik stand for?
  • Special Numbers
    problem
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    Special numbers

    Age
    11 to 14
    Challenge level
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    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?

  • Tilted Squares
    problem
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    Tilted squares

    Age
    11 to 14
    Challenge level
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    It's easy to work out the areas of most squares that we meet, but what if they were tilted?

  • Isosceles Triangles
    problem
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    Isosceles triangles

    Age
    11 to 14
    Challenge level
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    Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?