Magic Square Calculator

A normal magic square of order n is an n by n grid containing each integer from one through n squared exactly once. Its defining property is that every row, every column, and both long diagonals have the same total. That shared total is called the magic constant. Because all entries are known in advance, the constant can be calculated as n times the quantity n squared plus one, divided by two. For order three, the constant is fifteen; for order four, it is thirty-four. This calculator constructs a canonical square rather than searching randomly. Odd orders use the Siamese method. Place one at the top center, then move one row upward and one column right for each next number, wrapping around the grid edges. If that destination is occupied, move down one row instead. Repeating the rule fills every cell and automatically balances all required lines. The familiar order-three result begins with the row 8, 1, 6. Orders divisible by four use the doubly-even complement method. The grid is first filled in ordinary counting order. Entries in a symmetric pattern are then replaced by their complements relative to n squared plus one. This produces the classic order-four arrangement beginning 16, 2, 3, 13. Even orders that are not divisible by four are singly even. They are assembled from four related odd-order sub-squares and a carefully selected exchange of columns between quadrants. Magic squares have a long history in recreational mathematics, art, architecture, and cultural symbolism. Today they remain useful for teaching modular movement, symmetry, constructive algorithms, matrix indexing, and proof by invariant. They also make engaging number puzzles: remove several cells from a completed square, retain the target constant, and challenge a learner to reconstruct the missing entries. The generated square is deterministic, so entering the same order always returns the same valid arrangement. Many rotations, reflections, and other permutations of that arrangement are equally valid. The output limit of fifteen keeps the grid readable on a web page while still demonstrating all three construction families. To verify a result, add any displayed row and compare it with the magic constant, then repeat for a column and both diagonals. Every check should agree, and each integer in the required range should appear exactly once.