Conjecturing and generalising

  • Janine's Conjecture
    problem

    Janine's conjecture

    Age
    14 to 16
    Challenge level
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    Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. Does this always work? Can you prove or disprove this conjecture?
  • Expenses
    problem

    Expenses

    Age
    14 to 16
    Challenge level
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    What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
  • Legs Eleven
    problem

    Legs eleven

    Age
    11 to 14
    Challenge level
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    Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?
  • Dozens
    problem

    Dozens

    Age
    7 to 14
    Challenge level
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    Can you select the missing digit(s) to find the largest multiple?

  • Odd Differences
    problem

    Odd differences

    Age
    14 to 16
    Challenge level
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    The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
  • Hypotenuse Lattice points
    problem

    Hypotenuse lattice points

    Age
    14 to 16
    Challenge level
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    The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
  • Repeaters
    problem

    Repeaters

    Age
    11 to 14
    Challenge level
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    Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.
  • Card Trick 2
    problem

    Card trick 2

    Age
    11 to 14
    Challenge level
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    Can you explain how this card trick works?
  • Is there a theorem?
    problem

    Is there a theorem?

    Age
    11 to 14
    Challenge level
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    Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?
  • Equilateral Areas
    problem

    Equilateral areas

    Age
    14 to 16
    Challenge level
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    ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.