Magic Squares — Arrange Numbers So Every Line Adds the Same

A magic square is a grid where every row, every column and both diagonals add up to the same total. The oldest, the Chinese Lo Shu, is thousands of years old. You will build one with your hands.

Cut nine small squares of card and write the numbers 1 to 9, one on each tile. Draw an empty 3 by 3 grid to place them in.

Arrange the nine tiles in the grid so that every row, every column and both diagonals add up to 15. Hint: the numbers 1 to 9 add to 45, and 45 divided by 3 rows is 15, so 15 is the target. Another hint: 5 belongs in the middle. Keep swapping tiles until every line makes 15.

What you discovered. (1) For a square of side n using 1 to n-squared, the magic total is always n times (n-squared plus 1) over 2 -- 15 for a 3x3, 34 for a 4x4. (2) There is really only ONE 3x3 magic square; every solution you find is just that one rotated or flipped. (3) Bigger squares have their own tricks -- odd sizes can be filled by the 'always step up and to the right' Siamese method. (4) In 1514 the artist Albrecht Durer hid a 4x4 magic square in his engraving Melencolia I, with the year 1514 tucked into its bottom row -- proof that mathematicians and artists have loved these balanced numbers for centuries.