Conjecturing and generalising

  • Magic Circles
    problem

    Magic circles

    Age
    7 to 11
    Challenge level
    filled star filled star empty star
    Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?
  • Broken Toaster
    problem

    Broken toaster

    Age
    7 to 11
    Challenge level
    filled star empty star empty star
    Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
  • Dicing with numbers
    problem

    Dicing with numbers

    Age
    11 to 14
    Challenge level
    filled star empty star empty star
    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?
  • In a Spin
    problem

    In a spin

    Age
    14 to 16
    Challenge level
    filled star filled star filled star
    What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?
  • Overlap
    problem

    Overlap

    Age
    14 to 16
    Challenge level
    filled star filled star empty star

    A red square and a blue square overlap. Is the area of the overlap always the same?

  • Semi-Square
    problem

    Semi-square

    Age
    14 to 16
    Challenge level
    filled star filled star filled star
    What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
  • Square pizza
    problem

    Square pizza

    Age
    14 to 16
    Challenge level
    filled star filled star filled star
    Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?
  • A Tilted Square
    problem

    A tilted square

    Age
    14 to 16
    Challenge level
    filled star filled star filled star
    The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
  • Thank your Lucky Stars
    problem

    Thank your lucky stars

    Age
    14 to 16
    Challenge level
    filled star filled star empty star
    A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand corner of the grid?