Give every line the same total
A normal magic square uses every number from 1 to the number of cells exactly once. Every row, every column and both long corner-to-corner diagonals have the same total.
A 3 × 3 square uses 1 to 9 and each line totals 15. A 4 × 4 square uses 1 to 16 and totals 34. A 5 × 5 square uses 1 to 25 and totals 65. The practice shows the target total beside the board.
Start with a line that has one space. In the last row below, 2 + ? + 4 must equal 15, so the missing value is 15 − 2 − 4 = 9. Check its column as well: 1 + 5 + 9 = 15.
Use a crossing line when a row has two spaces
The first row of this 4 × 4 square has two spaces, so its total alone does not tell you each number. Look down the first column: ? + 12 + 2 + 15 = 34. The known numbers total 29, leaving 5.
Place 5 in the top-left cell. Now the top row has one space: 5 + ? + 16 + 10 = 34. Subtract the known total of 31 to get 3.
Check the second column: 3 + 14 + 8 + 9 = 34. Both inserted numbers are different, belong to 1 to 16, and agree with their crossing lines. Finish by checking every row, column and both long diagonals.
Find A before filling its neighbor
Each line in this square totals 34. The top row has spaces A and B. Find A using its column, which already contains 1, 11 and 6.
The two unused numbers are 10 and 16. Both belong somewhere in the square, so check the position instead of choosing a number merely because it is unused.
After finding A, use the top row to check B. These two choices check the method; the practice lets you place numbers directly in the board.
Try it
What belongs in A, above 1, 11 and 6 in the third column?
- 10
- 16
Show the answer
16
Correct. A = 34 − 1 − 11 − 6 = 16. The top row then gives 5 + 3 + 16 + B = 34, so B is 10.
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Practice Magic squares