Name Date
Magic square worksheets
Every row, column and diagonal adds to the same total. Fill in the missing numbers.
16
7
49
Total: 15
62
15
8
Total: 15
4
9
276
Total: 15
48
91
7
Total: 15
29
3
68
Total: 15
38
9
76
Total: 15
Magic square worksheets
Three by three squares where every row, column and diagonal adds to the same total. Choose how many cells are missing; every puzzle is checked to be solvable by reasoning alone.
Grade 2–Grade 6 · every refresh draws a new set · answer key included
Magic square worksheets — answer key
Every row, column and diagonal adds to the same total. Fill in the missing numbers.
816
357
492
Total: 15
672
159
834
Total: 15
438
951
276
Total: 15
438
951
276
Total: 15
294
753
618
Total: 15
438
951
276
Total: 15
6 problems · US Letter · answer key — 2 sheets, page breaks marked in the preview
What this worksheet is for
A magic square is the oldest recreational maths puzzle there is, and the three by three version has exactly one arrangement of the numbers one to nine — every square you will ever see is that one rotated or reflected. That makes it a perfect worksheet: the puzzle is always solvable, the arithmetic is addition a seven-year-old can do, and the reasoning is genuinely a puzzle. Squares here can also start from a number other than one and step by more than one, which changes every number on the page without changing the puzzle underneath.
Who it suits
Grade 2 to Grade 6, ages seven to twelve. One or two missing cells is a warm-up for Grade 2; four or five missing cells needs a strategy and suits Grade 4 upwards. Larger starting numbers and bigger steps make the addition harder without making the puzzle harder, which is a useful lever when the reasoning is fine and the arithmetic needs practice.
How to use the settings
Set the number of missing cells first — that is the difficulty. Then decide whether you want the classic one-to-nine square or a stretched version: a step of five starting at 10 gives a square of 10 to 50 with the same structure, which looks completely different to a child who has seen the standard one. The total is printed under each square, because without it the puzzles with many gaps would have no way in.
What goes wrong
Filling cells in reading order is the usual dead end: the top-left cell is rarely the one that can be worked out first. The habit to teach is scanning for a row, column or diagonal with only one gap, and it transfers to every other puzzle in this category. The second error is forgetting that the diagonals count too — a square that satisfies every row and column and fails a diagonal is not finished.
Questions
- What is the magic total?
- For the numbers one to nine it is 15. In general it is three times the middle number, which is why the middle cell of a three by three square is always the average — a useful thing to notice and a strong hint when solving.
- Can every puzzle be solved without guessing?
- Yes. Before a square is printed the generator checks that every missing cell can be deduced from a row, column or diagonal with one gap, working repeatedly until all gaps are filled. Arrangements that need trial and error are discarded.
- Are there other three by three magic squares?
- Not really — with the numbers one to nine there is only one, and the eight versions you see are its rotations and reflections. Changing the starting number and the step gives a different-looking square with the same underlying pattern.