Conjecturing and generalising

  • Special Sums and Products
    problem

    Special sums and products

    Age
    11 to 14
    Challenge level
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    Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
  • Sum Equals Product
    problem

    Sum equals product

    Age
    11 to 14
    Challenge level
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    The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 � 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?
  • Mini-max
    problem

    Mini-max

    Age
    11 to 14
    Challenge level
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    Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers and so on?
  • Plus Minus
    problem

    Plus minus

    Age
    14 to 16
    Challenge level
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    Can you explain the surprising results Jo found when she calculated the difference between square numbers?
  • Pinned Squares
    problem

    Pinned squares

    Age
    14 to 16
    Challenge level
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    What is the total number of squares that can be made on a 5 by 5 geoboard?
  • Lower Bound
    problem

    Lower bound

    Age
    14 to 16
    Challenge level
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    What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
  • Pareq Calc
    problem

    Pareq calc

    Age
    14 to 16
    Challenge level
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    Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel lines are 1 unit and 2 units.
  • Mindreader
    problem

    Mindreader

    Age
    11 to 14
    Challenge level
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    A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you the last two digits of her answer. Now you can really amaze her by giving the whole answer and the three consecutive numbers used at the start.
  • How old am I?
    problem

    How old am I?

    Age
    14 to 16
    Challenge level
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    In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

  • Differences
    problem

    Differences

    Age
    11 to 14
    Challenge level
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    Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?