Making Maths: Double-sided Magic Square

Age 7 to 14
Challenge Level

You may have seen magic squares before - if not have a look at problems from NRICH: Magic Squares, What's in a Name?. However this one is very special - it is double-sided.

This activity is based on an idea from the book "More Mathematical Curiosities" published by Tarquin - see below for details.

Follow the instructions below to make your square:

You will need:

  • Scissors
  • Clear plastic wallet (A4 or A5 size)
  • Plain thin card ( or A4 paper)
  • Ruler
  • Glue
  • Squared paper (optional)
  • Coloured pens or pencils (optional)

What to do:

  1. Cut down the plastic wallet so that it is approximately $15$cm by $15$cm (keep the edges sealed so that you still have a pocket).
  2. Now you need to make the numbered pieces. The easiest way to do this would be to print them out onto thin card or paper. Click here to see our versions. (Sixes and nines are underlined so that you can tell the difference between them.)
    If you can't print them, copy the numbers onto squared paper. Make sure you colour the squares as we have (or write the numbers in colour). It is important that the numbers written upside-down are upside-down!
  3. Cut out each double number set.
  4. Fold each pair of numbers away from you along the grey line and stick the backs together. You should end up with sixteen squares with a number on each side.

Before you try the double-sided puzzle, it is worth having a go at the magic square on just one side.
You need to arrange the numbers $1-16$ in a $4$ by $4$ grid so that:

  • All rows, columns and diagonals add up to $34$.
  • Each $2$ by $2$ group of the square adds up to $34$.

When you've done that (there are $48$ different ways of doing it!), you're ready to have a go at the Double-Sided Magic Square.

Slip the squares inside the plastic wallet so that you can see both sides.
The challenge now is to make sure both sides of the square obey the rules above.
The colours of the squares give you a clue to the arrangement:

  • On one side, all the numbers in the same row are the same colour. From top to bottom of the square, the rows are red, yellow, blue then green.
  • On the other side, all numbers in the same column are the same colour. From left to right these columns are blue, red, green and yellow.

To make a 3D version of the square, look at "More Mathematical Curiosities" published by Tarquin. It is priced at £4.95 and can be ordered directly from Tarquin Publications: