The Codebreaker's Grid

Magic sum for any 4×4 square using 1-16 is 34 (formula: 4×17/2). Dürer's square (16,3,2,13 / 5,10,11,8 / 9,6,7,12 / 4,15,14,1) is a verified magic square — all rows, columns, and both diagonals sum to 34. Rotating this square 90° produces a second valid magic square. There are 880 distinct 4×4 magic squares in total.

For any n×n magic square using consecutive integers 1 to n², the magic sum is always n(n²+1)/2. For n=4: 4×(16+1)/2 = 4×17/2 = 34. Verify the given square, then construct an alternative. Step 1: Magic sum formula: for integers 1 to n², magic sum = n(n²+1)/2 = 4×17/2 = 34. Step 2: Verify given grid rows: Row 1: 7+11+2+14=34. Step 3: Standard 4×4 magic square (Dürer's Melencolia I): Row 1: 16,3,2,13. Row 2: 5,10,11,8. Row 3: 9,6,7,12. Row 4: 4,15,14,1. Step 4: Verify Dürer's square — rows: 16+3+2+13=34. 5+10+11+8=34. 9+6+7+12=34. 4+15+14+1=34. Step 5: Columns: 16+5+9+4=34. 3+10+6+15=34. 2+11+7+14=34. 13+8+12+1=34. Step 6: Diagonals: 16+10+7+1=34. 13+11+6+4=34. This is a verified magic square. Rotate it 90° clockwise for a second valid square with sum 34. Key Insight: The magic sum formula n(n²+1)/2 comes from the fact that 1+2+...+n² = n²(n²+1)/2, and this total is distributed equally among n rows. There are exactly 880 distinct 4×4 magic squares using 1-16 — all with magic sum 34. Rotations and reflections of any one square give 7 more (8 total in its symmetry class).