Number Grids & Magic Squares

Magic Squares

In a magic square, every row, column, and diagonal adds to the same number — the magic sum.

Example 3×3 magic square (magic sum = 15):

Finding the Magic Sum

For a magic square using numbers 1 to n²:

• Sum of all numbers = n²(n² + 1) ÷ 2
• Magic sum = Total sum ÷ n
• 3×3: Total = 45 → Magic sum = 45 ÷ 3 = 15

In exam questions, just add any complete row or column to find the magic sum.

Grids with Arithmetic Rules

Some grids use operations across rows or columns.

Example: Each row multiplies to 24. Find the missing value (?):

Strategy & Key Tips

1. Find the magic sum (or rule) using a complete row or column.
2. For each incomplete row/column, subtract known values from the magic sum.
3. Verify that all rows, columns, and diagonals match.

Key Tips

• At least one complete row or column is always given — use it to establish the rule
• Diagonals must also equal the magic sum — use them to double-check your answers
• For non-magic-square grids, identify the operation (add, multiply) by testing a complete row
• Work from rows/columns with only one unknown value — solve these first