Magic Squares: History, Math and How to Make Your Own

Magic squares are grids of numbers in which every row, every column and both main diagonals add up to the same total. The classic version uses each whole number from 1 up to n × n exactly once, and its shared total, called the magic constant, is always n(n² + 1) ÷ 2. For a 3 × 3 square that's 15, for a 4 × 4 it's 34, and for a 5 × 5 it's 65.

This guide covers where magic squares come from (from the ancient Chinese Lo Shu to Albrecht Dürer's 1514 engraving and Benjamin Franklin's squares), the math behind them, and, best of all, step-by-step methods for building your own of any size. Every square in this article was checked by computer, so you can trust the numbers.

What Is a Magic Square?

A magic square of order n is an n-by-n grid of numbers where all the rows, all the columns and the two long diagonals have the same sum. Here's the smallest and most famous one, the 3 × 3 square known as the Lo Shu:

4	9	2
3	5	7
8	1	6

Check it: the rows are 4 + 9 + 2, 3 + 5 + 7 and 8 + 1 + 6, all 15. The columns are 4 + 3 + 8, 9 + 5 + 1 and 2 + 7 + 6, all 15. The diagonals are 4 + 5 + 6 and 2 + 5 + 8, both 15.

Useful terms

Wolfram MathWorld's entry on the magic square defines the vocabulary you'll see in puzzle books and math articles:

• Normal: the square uses the numbers 1 to n², each once. Most of this guide is about normal squares.
• Magic constant: the shared sum of every row, column and main diagonal.
• Semimagic: rows and columns work, but one or both main diagonals don't.
• Associative: every pair of numbers placed symmetrically opposite each other across the center adds up to n² + 1. In the Lo Shu, 4 and 6, 9 and 1, 2 and 8, and 3 and 7 all add to 10.
• Panmagic (also called pandiagonal or diabolic): even the "broken" diagonals that wrap around the edges add up to the magic constant.

The same entry notes that there's no magic square of order 2. Try it: in a 2 × 2 grid, the top row a + b and the left column a + c must match, so b = c, and a normal square can't repeat a number.

The Magic Constant Formula, Explained

You don't have to memorize the formula. You can work it out in two lines.

1. A normal square uses 1, 2, 3, … up to n². The sum of the whole numbers from 1 to m is m(m + 1) ÷ 2, so the total of every number in the square is n²(n² + 1) ÷ 2.
2. That total is split evenly across n rows, because every row has the same sum. Divide by n and you get the magic constant: n(n² + 1) ÷ 2.

For a 4 × 4 square: the numbers 1 to 16 add up to 16 × 17 ÷ 2 = 136, and 136 ÷ 4 = 34. Here are the first several orders:

Order (n)	Numbers used	Total of all numbers	Magic constant
3	1 to 9	45	15
4	1 to 16	136	34
5	1 to 25	325	65
6	1 to 36	666	111
7	1 to 49	1,225	175
8	1 to 64	2,080	260
9	1 to 81	3,321	369
10	1 to 100	5,050	505

The center of a 3 × 3 must be 5

Here's a neat bit of reasoning, also given on MathWorld's Lo Shu page. Four lines pass through the center: the middle row, the middle column and both diagonals. Each sums to 15, so together they total 60. Those four lines cover every cell of the grid once, except the center, which they count four times. So 60 = 45 + 3 × (center), which means the center is 15 ÷ 3 = 5.

And 9 can't go in a corner

A corner cell sits on three lines (a row, a column and a diagonal). If 9 were in a corner, each of those three lines would need two other numbers adding to 6. The only pairs from 1 to 8 that add to 6 are 1 + 5 and 2 + 4. The diagonal runs through the center, so it uses 5 and 1. That leaves only 2 + 4 for both the row and the column, and two lines can't share the same two numbers, so 9 has to sit in the middle of an edge, with 1 opposite it. Keep reasoning this way and you'll find the Lo Shu is the only 3 × 3 normal magic square, apart from its rotations and reflections. MathWorld confirms that it's unique and that it has eight arrangements. We checked all 362,880 orderings of 1 to 9 by computer and found exactly those eight.

A Short History of Magic Squares

The Lo Shu (China)

The Lo Shu is the oldest magic square on record. Chinese tradition tells of the pattern appearing on the shell of a turtle that came out of the Lo River, and a math page at Wichita State University, "Magic Squares," notes that the square may date back as early as 2200 B.C. That early date comes from legend rather than surviving documents, so treat it as tradition. What's certain is that the Lo Shu has been treated as a lucky symbol for a very long time.

India and the 4 × 4

The same Wichita State page says the earliest recorded fourth-order magic square is an inscription at Khajuraho, India, dated to roughly 1000 to 1100 A.D. Larger squares then spread through the Islamic world and into Europe, where they became objects of study for mathematicians and, sometimes, charms against bad luck.

Dürer's Melencolia I (1514)

The most famous magic square in art is in the upper right corner of Melencolia I, a 1514 engraving by the German artist Albrecht Dürer. The Metropolitan Museum of Art holds an impression (Melencolia I, Albrecht Dürer, 1514, engraving). Here's the square:

16	3	2	13
5	10	11	8
9	6	7	12
4	15	14	1

As MathWorld's page on Dürer's magic square points out, the two middle cells of the bottom row read 15 and 14, giving the year of the engraving. The square has far more than 34 in its rows, columns and diagonals. We verified each of these:

• Each 2 × 2 corner quadrant adds to 34: 16 + 3 + 5 + 10, 2 + 13 + 11 + 8, 9 + 6 + 4 + 15, and 7 + 12 + 14 + 1.
• The four center cells add to 34: 10 + 11 + 6 + 7.
• The four corners add to 34: 16 + 13 + 4 + 1.
• Any two numbers placed symmetrically across the center add to 17, so the square is associative.

Benjamin Franklin's squares (1750s)

Benjamin Franklin built magic squares to pass the time. MathWorld's page on the Franklin magic square dates his 8 × 8 square to 1750 and quotes his recollection that he made squares during long sessions of debate, when he was "tired with sitting there to hear debates." He described his squares in a 1752 letter to the English naturalist Peter Collinson, which the National Archives' Founders Online project publishes as "Benjamin Franklin to Peter Collinson, 1752"; its source note says the letter was printed in Franklin's Experiments and Observations on Electricity (London, 1769). Here's his best-known 8 × 8 square:

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

Strictly speaking it isn't a full magic square. Every row and column adds to 260, but the two main diagonals add to 228 and 292, which is why MathWorld calls it semimagic. Franklin cared more about other patterns, and they're remarkable. We checked all of these by computer:

• Every half row and every half column adds to 130.
• "Bent diagonals" add to 260. For example, going down from the top-left corner to the middle and back out to the bottom-left: 52 + 3 + 5 + 54 + 10 + 57 + 63 + 16 = 260.
• The four corners plus the four center cells add to 260: 52 + 45 + 16 + 17 + 54 + 43 + 10 + 23.
• Every 2 × 2 block of neighboring cells, anywhere in the square, adds to 130.

How Many Magic Squares Are There?

For normal squares, mathematicians count arrangements that aren't just rotations or reflections of each other (each square has eight versions that way). According to MathWorld:

Order	Distinct normal magic squares	Who counted it
3	1	The Lo Shu is unique
4	880	Frénicle de Bessy, 1693
5	275,305,224	R. Schroeppel, 1973
6	17,753,889,197,660,635,632	Mino, 2024 (earlier estimated by computer simulation at about 1.77 × 10 to the 19th power)

The jump is enormous. Going from 3 × 3 to 6 × 6 takes you from one square to about 17.75 quintillion. That's why there are simple, reliable recipes for building them: nobody finds a big magic square by trial and error.

How to Make an Odd Magic Square (3 × 3, 5 × 5, 7 × 7)

For any odd order, there's a beautifully simple recipe called the Siamese method, also known as the de la Loubère method. MathWorld gives the rule this way: start in the middle of the top row and keep moving up and to the right, wrapping around the edges. If the cell you'd move into is already filled, drop down one cell instead.

The rules in plain English

1. Put 1 in the middle cell of the top row.
2. For each next number, move one up and one right.
3. If that takes you off the top, wrap to the bottom row (same column). If it takes you off the right side, wrap to the left column (same row).
4. If the cell you land on is already filled (or you'd go off the top-right corner), instead put the number directly below the last number you wrote.
5. Keep going until the grid is full.

Worked example: a 5 × 5

Let's trace the first several numbers. Rows and columns are numbered 1 to 5 from the top-left.

• 1 goes in row 1, column 3 (the middle of the top row).
• Up-right from there is off the top, so wrap to the bottom: 2 goes in row 5, column 4.
• Up-right again: 3 goes in row 4, column 5.
• Up-right is off the right side, so wrap to the left: 4 goes in row 3, column 1.
• 5 goes in row 2, column 2.
• Up-right from 5 is row 1, column 3, but 1 is already there. So drop down instead: 6 goes directly below 5, in row 3, column 2.
• Carry on the same way: 7 in row 2, column 3; 8 in row 1, column 4; 9 wraps to row 5, column 5; and so on.

Here's the finished square. Every row, column and diagonal adds to 65.

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

Notice that the middle number of the range, 13, lands in the center, just as 5 did in the Lo Shu. Run the same method on a 3 × 3 and you get 8, 1, 6 / 3, 5, 7 / 4, 9, 2, which is the Lo Shu flipped top to bottom. It works for 7 × 7 (magic constant 175), 9 × 9 and every other odd size too. We tested it by computer through order 7.

How to Make a 4 × 4 or 8 × 8 Magic Square

Orders divisible by 4 (4, 8, 12 and so on) are called doubly even, and they have their own easy recipe. MathWorld describes it as filling the square in order and then replacing the entries on certain diagonals with their "complements."

The rules

1. Write the numbers 1 to n² in order, left to right and top to bottom.
2. Split the grid into 4 × 4 blocks, and in each block mark the cells on its two diagonals.
3. Replace every marked number a with (n² + 1) − a. Leave the others alone.

Worked example: a 4 × 4

Start with 1 to 16 in order. The marked cells are the two diagonals: 1, 6, 11, 16 and 4, 7, 10, 13. Each gets replaced by 17 minus itself: 1 becomes 16, 6 becomes 11, 11 becomes 6, 16 becomes 1, 4 becomes 13, 7 becomes 10, 10 becomes 7 and 13 becomes 4.

16	2	3	13
5	11	10	8
9	7	6	12
4	14	15	1

Every line adds to 34. Compare it with Dürer's square: swap the two middle columns of this one and you get Dürer's exactly. The swap happens to keep this square magic, which you can check on the diagonals: 16 + 10 + 7 + 1 and 13 + 11 + 6 + 4 both make 34.

The same recipe on an 8 × 8 (each marked number replaced by 65 minus itself) gives a square whose first row is 64, 2, 3, 61, 60, 6, 7, 57, and whose lines all add to 260. We checked the full grid by computer.

How to Make a 6 × 6 Magic Square (the LUX Method)

Orders like 6, 10 and 14 (even, but not divisible by 4) are called singly even, and they're the trickiest. MathWorld describes the mathematician John Conway's LUX method, which reuses the Siamese method on a smaller square.

The idea

For a 6 × 6, think of the grid as a 3 × 3 arrangement of 2 × 2 blocks. Each block gets a letter, L, U or X, and each letter is a pattern for filling its four cells with four consecutive numbers:

• L: top-right, bottom-left, bottom-right, top-left.
• U: top-left, bottom-left, bottom-right, top-right.
• X: top-left, bottom-right, bottom-left, top-right.

For a 6 × 6, the letters are: top row L L L, middle row L U L, bottom row U L U. (In general, the top rows are L's, the next row is U's, the rest are X's, and the middle U swaps places with the L above it. A 6 × 6 has no X rows.)

Filling it in

Now build a 3 × 3 Siamese square (8, 1, 6 / 3, 5, 7 / 4, 9, 2). Visit the blocks in that order. The block marked 1 gets the numbers 1 to 4, the block marked 2 gets 5 to 8, and so on, with block k getting 4k − 3 through 4k, placed in the order its letter says. Here's the result:

32	29	4	1	24	21
30	31	2	3	22	23
12	9	17	20	28	25
10	11	18	19	26	27
13	16	36	33	5	8
14	15	34	35	6	7

Every row, column and diagonal adds to 111. The same method produced a working 10 × 10 (magic constant 505) when we tested it.

Tricks: Changing a Magic Square Without Breaking It

Once you have one magic square, you can make many more. Each of these moves keeps every line equal:

Change	What happens to the magic constant	Example with the Lo Shu
Add the same number k to every cell	Goes up by n × k	Add 10: constant becomes 15 + 30 = 45
Multiply every cell by m	Is multiplied by m	Double: constant becomes 30
Replace each a with (n² + 1) − a	Stays the same	4 becomes 6, 9 becomes 1, and so on
Rotate or reflect the square	Stays the same	Gives the 8 versions of the Lo Shu

Make a 4 × 4 square for any total

Want a square whose lines all add to a birthday number, a year or a team score? Here's a method we built and verified. Start with this normal 4 × 4 magic square, in which the four biggest numbers (13, 14, 15 and 16) sit one per row, one per column, and one on each main diagonal:

1	5	12	16
15	11	6	2
10	4	13	7
8	14	3	9

To reach a target total T of 34 or more:

1. Work out T − 34 and divide by 4. Call the whole-number answer k and the remainder r (0, 1, 2 or 3).
2. Add k to every cell.
3. Add r more to the four cells that started as 13, 14, 15 and 16.

Because those four cells hit every row, column and diagonal exactly once, each line goes up by exactly 4k + r, which is T − 34. For a target of 100: 100 − 34 = 66, and 66 ÷ 4 is 16 remainder 2. Add 16 everywhere, then 2 more to the big four, and you get 17, 21, 28, 34 / 33, 27, 22, 18 / 26, 20, 31, 23 / 24, 32, 19, 25. Every line adds to 100, and all sixteen numbers are different.

Panmagic Squares: When Even the Broken Diagonals Work

A panmagic square goes one better than an ordinary magic square. Imagine the square printed on a tile and repeated across a floor. In a panmagic square, any diagonal you trace across that floor adds to the magic constant, not just the two long diagonals. On a single square, those extra lines show up as "broken diagonals" that wrap from one edge to the opposite edge.

Here's a 4 × 4 panmagic square that we checked line by line:

1	8	13	12
14	11	2	7
4	5	16	9
15	10	3	6

Take the broken diagonal that starts at 8 in the top row and runs down to the right: 8, 2, 9 and then wraps to the left edge for 15. That's 8 + 2 + 9 + 15 = 34. All four down-right diagonals and all four down-left diagonals add to 34 the same way. This square has another bonus: every 2 × 2 block of neighboring cells adds to 34, even blocks that wrap around the edges, like the four corners (1 + 12 + 15 + 6).

Panmagic squares have a fun party trick. Because every diagonal works on the "tiled floor," you can move the top row to the bottom, or the left column to the right, and the result is still panmagic. One square gives you a whole family of new ones.

Which squares can be panmagic?

The Lo Shu can't. MathWorld's Lo Shu page notes that it's associative but not panmagic, and you can see why: its broken diagonal 9, 7, 8 adds to 24, not 15. And notice that the panmagic square above isn't associative. Pairs across its center, like 1 and 6, don't add to 17. Dürer's square is the reverse: it's associative but not panmagic. Different squares have different extra powers, which is part of what keeps mathematicians counting and classifying them.

Magic Squares, Latin Squares and Sudoku Compared

Magic squares often get mixed up with other number grids. Here's how they differ.

Grid type	The rule	Numbers used	Example size
Magic square	Every row, column and main diagonal has the same sum	Usually 1 to n², each once	3 × 3 using 1 to 9
Latin square	Each symbol appears once in every row and column	n symbols, each used n times	4 × 4 using 1 to 4
Sudoku	A Latin square where each box also has every symbol once	1 to 9, each used nine times	9 × 9
Kakuro and killer sudoku	Groups of cells must reach given sums with no repeats	1 to 9	Varies

The key difference is that a magic square is about sums, while a Latin square or sudoku is about placement. A completed sudoku does have equal sums in every row and column (each adds to 45), but that's a side effect, not the rule. Killer sudoku and kakuro mix the two ideas, which is why solvers of those puzzles keep sum tables handy.

Common Mistakes When Building Magic Squares

• Going the wrong way in the Siamese method. It's up and to the right. If you start going down, you'll still get a magic square only if you consistently mirror every rule, so pick one direction and stick with it.
• Forgetting the "drop down" rule. When the next cell is taken, the new number goes directly below the last number you wrote, not below the blocked cell.
• Using the odd-order method on an even square. The Siamese method only works for odd orders. Use the diagonal-complement method for 4, 8, 12 and the LUX method for 6, 10, 14.
• Checking only rows. A grid with equal rows and columns but unequal diagonals is semimagic, like Franklin's 8 × 8. Always add both diagonals.
• Miscounting the constant. Use n(n² + 1) ÷ 2 before you start, so you know what every line should total.

Try It Yourself

Puzzle 1: Fill the gaps

This is Dürer's square with five numbers missing. Each line adds to 34. Can you fill them in?

16	3	?	13
5	?	11	8
?	6	7	12
4	15	?	?

Hint: start with the top row, then the second row, then the left column.

Answer: top row 34 − 16 − 3 − 13 = 2; second row 34 − 5 − 11 − 8 = 10; left column 34 − 16 − 5 − 4 = 9; third column 34 − 2 − 11 − 7 = 14; and the bottom row 34 − 4 − 15 − 14 = 1.

Puzzle 2: Build a 7 × 7

Use the Siamese method to build a 7 × 7 square. Its magic constant should be 7 × 50 ÷ 2 = 175, and 25 (the middle of 1 to 49) should land in the center. If your middle row adds to 175 and the center is 25, you're almost certainly right.

Puzzle 3: A square for your own number

Use the "any total" method from the tricks section to build a 4 × 4 square whose lines all add to 50. Work it out on paper before you peek.

Answer: 50 − 34 = 16, and 16 ÷ 4 is 4 with no remainder. So add 4 to every cell of the starting square and you're done: 5, 9, 16, 20 / 19, 15, 10, 6 / 14, 8, 17, 11 / 12, 18, 7, 13. Check a row (5 + 9 + 16 + 20 = 50), a column (5 + 19 + 14 + 12 = 50) and a diagonal (5 + 15 + 17 + 13 = 50). Now try your own birthday year, house number or favorite number. Any total from 34 up works.

How to check any square quickly

Checking is half the fun, and it's where most mistakes get caught. Here's a fast routine:

1. Work out the target first with n(n² + 1) ÷ 2, or, for a non-normal square, add up the first row and use that.
2. Add every row. Write each total in the margin.
3. Add every column. Write each total below the grid.
4. Add both long diagonals.
5. For a normal square, make sure no number repeats and none is missing. A square with a repeated number can still have equal lines, but it isn't a normal magic square.

For a 4 × 4 that's ten sums, and for a 5 × 5 it's twelve. If one line is off, look for the cell that sits on both that line and another line that's off, since a single wrong number throws off two or three lines at once.

Magic Squares in Puzzles and Classrooms

Magic squares sit right at the meeting point of arithmetic and logic, which is why they show up in so many number puzzles. Sudoku's rule that each digit appears once per row, column and box comes from a related family of grids. If you like that connection, our guide to the mathematics of sudoku goes deeper, and where sudoku came from tells the story of how number grids became a puzzle craze. Sum-based logic is also the heart of killer sudoku combinations, where you work out which digits can make a given total.

For teaching, a magic square is a sneaky way to get dozens of addition facts practiced in one sitting, and the "fill the gaps" style above works for many ages. Our math crossword page and puzzles for homeschool have more ideas, and if you'd like more number-placement puzzles to try, see games like sudoku.

Wrap-Up: Magic Squares and Your Next Puzzle

Magic squares have fascinated people for centuries because they're simple to check and surprisingly deep. Remember the formula n(n² + 1) ÷ 2, the up-and-right Siamese method for odd sizes, the diagonal-complement trick for multiples of 4, and the LUX method for the in-between sizes, and you can build a magic square of any order except 2. If you're putting together a puzzle packet to go with one, our free puzzle maker lets you type your own words or pick a theme, choose a puzzle type, and print it with an answer key.

Four Building Methods Compared

Different sizes of magic square call for different recipes. This table compares the four methods in the guide by the sizes they fit and the key step.

Method	Sizes it fits	The key step
Siamese method	Odd orders such as 3 × 3, 5 × 5 and 7 × 7	Start in the middle of the top row and keep moving up and to the right, wrapping around the edges and dropping down one cell when the next cell is filled
Complement method	Orders divisible by 4, such as 4 × 4 and 8 × 8	Write the numbers in order, mark the diagonals of each 4 × 4 block and replace each marked number a with (n² + 1) − a
LUX method	Orders like 6, 10 and 14 that are even but not divisible by 4	Treat the grid as 2 × 2 blocks marked L, U or X and visit them in the order of a smaller Siamese square
Target-total method	A 4 × 4 square for any total of 34 or more	Add k to every cell, then add r more to the four cells that started as 13, 14, 15 and 16

Frequently Asked Questions

What is the formula for the magic constant?

For a normal magic square of order n, using the numbers 1 to n squared once each, the magic constant is n times (n squared plus 1), divided by 2. The reason is simple: all the numbers together add up to n squared times (n squared plus 1) divided by 2, and that total is shared equally among n rows. So a 3 by 3 square has a constant of 15, a 4 by 4 has 34, a 5 by 5 has 65, a 6 by 6 has 111 and an 8 by 8 has 260.

Why is 5 always in the center of a 3 by 3 magic square?

Four lines pass through the center cell: the middle row, the middle column and the two diagonals. Each adds to 15, so together they total 60. Those four lines include every cell of the grid once, plus the center three extra times. The whole grid adds to 45, so 60 equals 45 plus three times the center, and the center must be 5. This is also why the Lo Shu is the only 3 by 3 normal magic square, apart from its eight rotations and reflections.

Is there a 2 by 2 magic square?

No, not a normal one. In a 2 by 2 grid, the top row is a plus b and the left column is a plus c. For those to match, b must equal c, and a normal magic square can't use the same number twice. The same problem appears with the other lines, so every number would have to be equal. Magic squares exist for every other order from 3 upward, and there are standard construction methods for odd orders, for multiples of 4, and for the remaining even orders like 6 and 10.

How do you make a 3 by 3 magic square?

Use the Siamese method. Put 1 in the middle of the top row. For each next number, move one cell up and one cell right, wrapping around to the bottom or the left side if you go off the grid. If the cell you'd move into is already filled, put the number directly below the last one instead. Following these steps gives 8, 1, 6 on top, 3, 5, 7 in the middle and 4, 9, 2 on the bottom. Every row, column and diagonal adds to 15.

What is the magic square in Durer's Melencolia I?

Albrecht Durer's 1514 engraving Melencolia I includes a 4 by 4 magic square in its upper right corner. The rows read 16, 3, 2, 13; 5, 10, 11, 8; 9, 6, 7, 12; and 4, 15, 14, 1. Every row, column and diagonal adds to 34, and so do the four corners, the four center cells and each 2 by 2 quadrant. The two middle numbers of the bottom row, 15 and 14, spell out the year the engraving was made, which is a neat bit of hidden signature.

Did Benjamin Franklin make magic squares?

Yes. Franklin built magic squares to pass the time during long debates, and he described them in a 1752 letter to Peter Collinson, later printed in his book Experiments and Observations on Electricity. His famous 8 by 8 square has rows and columns adding to 260, half rows and half columns adding to 130, and bent diagonals adding to 260. Its two main diagonals don't add to 260, so mathematicians call it semimagic. Franklin was more interested in the extra patterns, which are remarkably rich.

How many 4 by 4 magic squares are there?

There are 880 distinct normal 4 by 4 magic squares, not counting rotations and reflections as different. That count was worked out by the French mathematician Bernard Frenicle de Bessy in 1693. Since each square can be rotated and reflected into eight versions, that's 7,040 arrangements if you count every orientation separately. For comparison, there's only one 3 by 3 square and about 275 million 5 by 5 squares, which shows how quickly the numbers grow as squares get bigger.

Can I make a magic square that adds up to any number?

Yes, with a 4 by 4 square you can reach any total of 34 or more. Start with a normal 4 by 4 magic square in which 13, 14, 15 and 16 sit one per row, one per column and one on each diagonal. Subtract 34 from your target and divide by 4. Add the whole-number answer to every cell, then add the remainder to the four cells that started as 13 to 16. Each line rises by the same amount, so every line hits your target, and the numbers stay different.

Are magic squares good for teaching math?

They're a great way to practice addition and mental math because checking a single square means adding many lines, and filling gaps in a partly finished square turns that practice into a puzzle. Older students can explore why the magic constant formula works, why the center of a 3 by 3 must be 5, or how the construction methods keep every line equal. Those questions lead naturally into algebra and proof. Start younger learners with the 3 by 3 Lo Shu and fill-the-gap puzzles.

Sources

We checked every source below while writing this guide. Last reviewed October 30, 2026, by Joan Ramos.

1. Magic Square (Wolfram MathWorld)
2. Lo Shu (Wolfram MathWorld)
3. Duerer's Magic Square (Wolfram MathWorld)
4. Franklin Magic Square (Wolfram MathWorld)
5. Melencolia I, Albrecht Durer, 1514 (The Metropolitan Museum of Art)
6. Benjamin Franklin to Peter Collinson, 1752 (Founders Online, National Archives)
7. Magic Squares (Wichita State University, Department of Mathematics)