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Magic Squares
Fill in the missing cells so every row, column, and diagonal sums to the same value S.
6 puzzles where arithmetic is enough — no equations needed. Look for a complete row, column, or diagonal to find S, then fill in the blanks.
8 puzzles with integer entries where algebra is required. The four given values don't let you find S directly — you'll need to set up equations.
6 puzzles with mixed fractions and integers. Same technique as Advanced — enter fractions as n/d (e.g., 22/3).
When do you need equations?
- Given the 4 edge midpoints: pair sums cancel S from the diagonals
- Given the 4 corners: both diagonals pass through the center — locking c to S − (pair sum)
- Both configurations require a system of 2+ equations to find S
Edge Midpoints — Integers (S = 15)
The four edge midpoints are 2, 4, 6, and 8. Learn how the diagonal trick eliminates S and lets you solve for all 5 unknowns using two equations.
Corners — Integers (S = 21)
The four corner values are 10, 9, 5, and 4. Both diagonals through the center give the same equation for c — so Column 2 alone determines S.
Edge Midpoints — Mixed Fractions (S = 24)
Two edge midpoints are integers (10 and 6), two are sevenths (38/7 and 74/7). The algebra is identical to the integer case — fractions just tag along for the ride.
