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Term

Magic Constant

The common row, column, and diagonal sum of a normal magic square of order k ≥ 3: k(k²+1)/2 → 15, 34, 65, 111, ….

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The magic constant (or magic sum) is the number that every row, column, and main diagonal of a normal magic square of order \(k\) must add up to: \(M_k = k(k^2+1)/2\). For the familiar 3×3 square it's 15; for 4×4 it's 34, for 5×5 it's 65, then 111, 175, 260, 369.

The most famous is the order-4 magic constant 34, immortalised in Albrecht Dürer's 1514 engraving Melencolia I, whose 4×4 square sums to 34 every way — including the four quadrants and the four centre cells — and encodes the date 1514 in its bottom row. The order-3 square (summing to 15) is the unique 3×3 magic square up to rotation, the Lo Shu square of Chinese legend.

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