Magic Squares and Sum Sequences

A magic square has the same total along every row, every column and both diagonals. The first move is always the same: find a complete line and add it up. Here the top row is complete, twenty plus fifteen plus sixteen is fifty-one, so every line adds to fifty-one. Then any line with two numbers and one blank gives the blank at once: fifty-one minus the two numbers. The left column gives thirteen for the middle-left cell. But the question wants the bottom-right corner, and the fastest path to it runs through the centre along a diagonal — which is the next step.

The centre sits on four lines — its row, its column and both diagonals — which makes it the most useful cell in the square. The diagonal from eighteen to sixteen has one blank, the centre: fifty-one minus eighteen minus sixteen is seventeen. Now the other diagonal, from twenty through the centre to the corner, also has one blank: fifty-one minus twenty minus seventeen is fourteen. Two lines and the corner is found, without touching the rest. And notice something about that centre: three times seventeen is fifty-one. In a three-by-three magic square the total is always three times the centre.

The classic magic square uses the numbers one to nine. They add to forty-five, spread over three rows, so each line adds to fifteen, and the centre is five. Look at the pairs opposite each other through the centre: eight and two, one and nine, six and four, three and seven. Every pair adds to ten, twice the centre. That is a rule you can carry to any three-by-three magic square. The total is three times the centre; any two cells opposite each other through the centre add to twice the centre. With a centre of seventeen and a corner of twenty, the opposite corner is thirty-four minus twenty, fourteen — no lines needed.

A sum sequence makes each term from the ones before it. Three, five, eight, thirteen, twenty-one: each term is the previous two added. One, two, three, six, eleven, twenty: each is the previous three added. Check the rule on the terms you are given before you extend it — is it two or three, and is it addition? Then it runs in either direction. Forwards, add. Backwards, subtract: if something, something, seven, eleven, eighteen is a previous-two sequence, the term before seven must be eleven minus seven, four, and the one before that seven minus four, three.

The hardest version puts unknowns inside the sequence. Two, X, zero, Y, six, Z, where each term is the sum of the previous three. Write the rule wherever you can: Y is two plus X plus zero, and six is X plus zero plus Y. So Y is X plus two, and X plus Y is six — two unknowns, two facts, exactly the last lesson. X is two, Y is four, and then Z is zero plus four plus six, ten. Check the whole sequence against the rule. Guess-and-check works as well: try X equals one and the fifth term comes out four instead of six, so X must be bigger. Quiz and exercises now.