Printable magic square worksheets
Choose the size and the number of blanks, and the page makes a sheet of magic square puzzles with an answer page. Every puzzle has exactly one solution, and the sheet number lets you print the same sheet again.
Clues per puzzle
given numbers; the rest are blank
Sheet number: . The sheet and the answer page are below.
How this is calculated
Every sheet prints the magic constant, which for a normal square of order n is
M = n(n² + 1) / 2
That is 15 for 3 × 3 (3 × 10 / 2) and 34 for 4 × 4 (4 × 17 / 2). Clues per puzzle is n² minus the blanks you choose: for a 3 × 3 sheet with 4 blanks, 9 − 4 = 5 clues.
Each puzzle starts from a magic square, rotated or reflected at random. The blank cells are then chosen so that their rows and columns let you work out every missing number, and the page confirms with the solver that exactly one completion exists. Each puzzle on a sheet uses the sheet number plus its position as its seed, so entering the same sheet number gives the same sheet again.
Limits
- Orders 3 and 4. The 3 × 3 puzzles come from the 8 rotations and reflections of one square, so a small sheet can contain a repeat. The 4 × 4 puzzles come from a small set of squares and their rotations, reflections and complements (each number v replaced by 17 − v).
- Blanks: 1 to 7 per puzzle. Zero is rejected because a puzzle needs a blank, and more than 7 can leave several answers.
- Puzzles per sheet: 1 to 6. Paper size changes only the page setup used when you print.
- Difficulty is simply the number of blanks you pick. The page does not label puzzles easy or hard.
Sources
- Weisstein, Eric W. “Magic Square.” MathWorld, A Wolfram Resource. mathworld.wolfram.com/MagicSquare.html (constant formula).
Using the sheets in class
Print one sheet per student or pair and keep the answer page for checking. Printing the answer page separately works too: choose its page in the print dialog. Because the sheet number is printed on every page, a teacher can regenerate the same sheet later.
A strategy handout can be as short as three lines: work out the constant, look for a line with exactly one blank and fill it, repeat. The 3 × 3 guide and the 4 × 4 guide explain the reasoning in more detail. To try puzzles on screen first, use the magic square puzzle. A stuck grid can go to the solver.
Back to the magic square generator.