WE’RE NEARING the end of Maths Week, which began last Saturday… let us know in the comments how you’ve been doing with our puzzles.

Day eight – Magic squares

The classic 3 by 3 magic square arranges the numbers 1, 2, 3, … 9 in a grid such that all row and column sums (and the two diagonals) are the same. They should sum to 15.

Can you complete this square?

Such arrangements have fascinated people since ancient times. They were known in many ancient cultures, including in China about 200 BCE.

You can rotate such a square, or flip it vertically or horizontally, to get “new” magic squares. But, there are no other possibilities using the numbers 1, 2,3, … 9. Only 5 can be in the middle, with the other numbers cycling around it in a prescribed order.

2) Is it possible to arrange the numbers 1, 2, 3, 4 in a 2 by 2 square with a similar property?

3) Can you find nine different whole numbers (no two the same), and arrange them in a square with each row, column, and diagonal sum being 15 as before, with 8 in the middle of the top row? You will need to use at least one number larger than 9.

4) The idea of using consecutive whole numbers can be extended: let’s use more of them. Can 1, 2, 3, … 16, be arranged in a 4 by 4 square so that all rows, column and diagonal sums are the same?

There are many ways to do this, and we are just asking can you figure out what the common row, column and diagonal total must be? Is it bigger than 20? Than 40?

Such 4 by 4 magic squares were known in India by 600 CE. It turns out that there are 880 fundamentally different ways to do this. In Europe, there’s an example in Albrecht Dürer’s painting Melancholia from 1514. This idea can be extended to larger squares, and this was done by Chinese, Indian, and Arab scholars over 500 years ago. US President Ben Franklin explored 8 by 8 magic squares in the 1750s.

Let’s return to 3 by 3 magic squares, but make it a bit more challenging: let’s switch is from addition to multiplication.

5) Find nine different positive whole numbers (each less than 50, say) and form a 3 by 3 grid of them such that the product of the three numbers in any row, column or diagonal, when we multiply them always comes out the same. The lowest possible magic constant for such a square is 216.

Here is a first attempt at this:

The row products are all 32, as are the column products, but the diagonal products are 4 and 8. Also, numbers are repeated.

Can you do better?

Friday’s puzzle: The answer

1) 32. The most you could take out of just one colour is 30. You would have 15 pairs. Then
drawing out two more would add a pair of the other colour.

2) 3. IF you don’t get a pair from the first two you will have one of each. The next sock will match one of those.

3) 4. Your first three draws could give you one sock of each colour. Then the next sock will match one of those.

4) The most you can have without a white sock is 24 (13 black and 11 blue). Then the next two socks will have to be a white pair.

5) Choose from the drawer marked “Mixed”. Say you pick a black sock: you know that this must be the black sock drawer. (It can’t be “Mixed”). As all the drawers are wrongly labelled, the drawer marked “White” must then be the mixed sock drawer. Therefore, the drawer marked Black must be the white sock drawer.

6) 3 that is 1 blue, 1 green and 1 white.

Come back tomorrow at 7.30pm for the answers to today’s puzzle. 

The puzzles this week have been compiled by Eoin Gill and Colm Mulcahy of Maths Week Ireland / South-East Technological University (SETU).