The MAGIC OF MATHS book tells you all about magic squares, and How to Make 4x4 Magic Squares which will produce any number. Here are how some bigger squares work.
This is the basic 5x5 magic square.
It uses all the numbers 1-25 and it adds up to 65 in 13 different ways:
This square adds up to 62 in 13 ways.
You'll see it's very similar to the first square but we've subtracted 3 from each number in a red box. That's why each line adds up to 3 less than 65.
How to lay out a 5x5 Magic Square
Have another look at the way the numbers are set out in the original square. It uses all the numbers 1-25, and if you follow the numbers round in order you'll see they appear in this pattern:
1,2,3,4 and 5 are in a diagonal line, which goes off the top and comes back at the bottom, then goes off the right and comes back on the left. Once the first five numbers are in place, there's no empty place to put number 6.
Again you'll see that the numbers 6-10 are in a diagonal which goes round until there's no space for the 11. So the 11 goes under the last number which was 10. If you keep going, you'll fill the whole grid with the numbers 1-25 and make the basic 5x5 magic square.
** You could start with the number 1 anywhere, but if you put it in the middle of the top line, this ensures that the diagonals work and that the 4 corners and the middle number add up to 65.**
Instead of starting with the number 1, start with a 4, then continue filling in 5,6,7,8 etc.
until you finish on 28.
In this case it's 4 x 5 + 60 = 20 + 60 = 80
A 7x7 square works the same way as a 5x5 square - just fill in the numbers in diagonals as before. Sadly the four corners and middle number don't give the right result, but you'll find all the lines and diagonals add up to 175!
The classic "Knight's Puzzle" is to try and move a knight round a chess board visiting every square just once. It's a tough puzzle at the best of times, but here is one very special solution! The knight starts on the square numbered 1 then hops to 2, then 3 etc. finally finishing on 64. (It could then hop back to 1 and start again!)
Here's the good bit - every row and every column add up to 260!
Finally look at this peculiar 4x4 magic square.
Every row and column and both diagonals add up to 19,998 - but if you turn your computer screen upside down it still works!
There's a reasonably simple explanation for this. Can you see it?