How to make bigger Magic Squares

5x5 Magic Squares   

7x7 Magic Squares   

The 8x8 "Knight's move" Magic Square   

An Upside Down Magic Square

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.

5x5 Magic Squares

17	24	1	8	15
23	5	7	14	16
4	6	13	20	22
10	12	19	21	3
11	18	25	2	9

This is the basic 5x5 magic square.

It uses all the numbers 1-25 and it adds up to 65 in 13 different ways:

• All 5 horizontal lines add up to 65
• All 5 vertical lines add up to 65
• The two diagonals both add up to 65
• Finally you can add up the four corners and the number in the middle to get 65.

How to make a 5x5 magic square add up to other numbers.

17	24	1	5	15
20	5	7	14	16
4	6	10	20	22
10	12	19	21	0
11	15	25	2	9

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.

If you wanted each line to add up to 80, that's 15 more than 65. So starting with the original square, you'd just add 15 to each number in a red square. However, we can do better than that!

How to lay out a 5x5 Magic Square

		1
	5
4	6
				3
			2

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.

		1	8
	5	7
4	6
10				3
11			2	9

The rule is to put the 6 UNDER the 5, and then continue putting numbers in another diagonal line:

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.**

So suppose you want the square to add to 80?

20	27	4	11	18
26	8	10	17	19
7	9	16	23	25
13	15	22	24	6
14	21	28	5	12

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.

You get a square like this one:

You can always work out what the lines of a 5x5 square like this will add up to:

• Take the lowest number and multiply by 5.
• Add 60

In this case it's 4 x 5 + 60 = 20 + 60 = 80

7x7 Magic Squares

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!

30	39	48	1	10	19	28
38	47	7	9	18	27	29
46	6	8	17	26	35	37
5	14	16	25	34	36	45
13	15	24	33	42	44	4
21	23	32	41	43	3	12
22	31	40	49	2	11	20

The "KNIGHT'S MOVE" 8x8 Magic Square

If you know the rules of chess, you'll know that it's played on a board measuring 8 squares by 8 squares. One chess piece is the "knight" and it moves around the board in a strange way. It leaps around in a little "L" shape - in other words it goes two steps forward and one step to the side. In this little diagram, if the "knight" was on the yellow square in the middle, then the red squares show where it could jump to.

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!)

50	11	24	63	14	37	26	35
23	62	51	12	25	34	15	38
10	49	64	21	40	13	36	27
61	22	9	52	33	28	39	16
48	7	60	1	20	41	54	29
59	4	45	8	53	32	17	42
6	47	2	57	44	19	30	55
3	58	5	46	31	56	43	18

Here's the good bit - every row and every column add up to 260!

The "UPSIDE DOWN" Magic Square

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?

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