Squares

  • Coloured Edges
    problem

    Coloured edges

    Age
    11 to 14
    Challenge level
    filled star empty star empty star
    The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
  • Square Corners
    problem

    Square corners

    Age
    7 to 11
    Challenge level
    filled star empty star empty star

    What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

  • Rope Mat
    problem

    Rope mat

    Age
    7 to 11
    Challenge level
    filled star filled star empty star
    How many centimetres of rope will I need to make another mat just like the one I have here?
  • Take a square
    problem

    Take a square

    Age
    14 to 16
    Challenge level
    filled star filled star filled star
    Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another 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?
  • Similarly so
    problem

    Similarly so

    Age
    14 to 16
    Challenge level
    filled star filled star filled star
    ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.
  • Tetra Square
    problem

    Tetra square

    Age
    14 to 18
    Challenge level
    filled star filled star empty star
    ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.
  • 2001 Spatial Oddity
    problem

    2001 spatial oddity

    Age
    11 to 14
    Challenge level
    filled star filled star filled star
    With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
  • Folding Squares
    problem

    Folding squares

    Age
    14 to 16
    Challenge level
    filled star filled star empty star
    The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?