Math & Logic

Testing a rule on a pair you have not used is a good check. Here is a sequence that passes it five times over and still has a different rule underneath.

Sudoku has nothing to do with adding up. It is a puzzle about where things cannot go, until only one possibility is left.

Puzzles come in kinds, and each kind is beaten a different way. Knowing which one you are holding is most of the battle.

Your job is not really to find the missing number. It is to find the rule. Once you have the rule, the number falls out on its own.

A logic grid puzzle never tells you an answer directly. You get there by ruling things out, until exactly one possibility is left standing.

Every row, column and diagonal has to add up to 15. Not because somebody chose 15, but because it is the only total the puzzle can have.

The first sudoku move needs a row that is nearly full. Here is the one that works when nothing is nearly full, and it asks a different question.

Nobody puts the five in the middle. Counting forces it there — and forces every other number too, which is why there is really only one magic square.

One rule makes a list of numbers that turns up in pinecones and sunflowers. Build the spiral yourself and find out why plants land on it.

You will know their answer before they have finished. You never learn their number, never see their working, and it works every time.

Your fingers can give you every answer in the 9-times table. Hold up both hands, and there is nothing left to memorize.

Change the 8 and you get a new trick. Change a different step and it quietly stops being one. Two running counts tell you which is which before you test it.

Nobody chose 137.5 degrees. It is what you get by dividing a circle by the golden ratio, and the sequence has been creeping toward that number all along.

The finger trick runs out at ten fingers. The rule underneath it never runs out, because every ten, hundred and thousand is a pile of nines plus one.

A team scores 3, then 1, then 4, then 2, then 5. Are they any good? An average squashes five numbers into one you can actually compare.

This page picks a secret number and tells you higher or lower. There is a way to win every time, and it never needs more than seven guesses.

If it is raining, then I will take my umbrella. You say sentences like that without thinking. Computers and detective stories run on the same shape.

Twenty sums on a worksheet is a chore. Twenty sums to get your counter round the board first is not, and it is the same twenty sums.

Four legs does not prove a dragon is blue. Two legs proves it is not, every time — and that backwards step is the one detectives actually use.

Halving finds any number from 1 to 100 in seven. Showing that nobody could ever do it in six is a different job, and it needs a different kind of counting.

Shift every letter by the same amount and A becomes D. Julius Caesar sent orders this way, and the shift number is the whole secret.

The last digit under a barcode is not part of the number. It is a guard, worked out from the other twelve, and it catches almost every slip.

A treasure hunt where each answer is also a direction. Solve the sum, and the sum tells you where the next clue is hidden.

Count a solid three ways and combine them, and every solid you can find gives the same answer. One drawing on a balloon shows why it has no choice.

Twelve tiles cover the same floor however you arrange them. The line round the outside is not the same at all, and one rule predicts it exactly.

Draw any map you like and colour it so no two neighbours match. Three colours will fail. Proving four always works took people 124 years.