Follow the Logic

What deduction actually promises

A deductive puzzle hands you a set of rules (premises) and asks for a conclusion that must be true if the rules are. That word "must" is the whole pleasure. You're not guessing what's likely; you're proving what's unavoidable. If an answer is merely plausible, you haven't finished.

The hidden constraint in most logic puzzles is a rule you read but didn't use. Puzzle setters rarely include decoration. When a clue seems pointless, it's usually the one doing the heavy lifting, and noticing it is the moment the puzzle opens.

Knights, knaves and the case split

Raymond Smullyan popularised a family of puzzles in What Is the Name of This Book? (1978): an island where knights always tell the truth and knaves always lie. The method that cracks nearly all of them is the case split. Assume one possibility, follow it to the end, and see whether it explodes.

Worked example

A says: "B is a knave." B says: "A and I are the same type." Suppose A is a knight. Then B really is a knave, so B's statement is a lie, meaning A and B are different types. A knight and a knave are different, so that's consistent. Now suppose A is a knave. Then B is actually a knight, so B's claim is true: they're the same type. But a knave and a knight aren't the same. Contradiction. So A is a knight and B is a knave.

Notice what we did: we didn't need to be clever, just thorough. Two cases, each checked to the end. Patience beats brilliance here.

Why "if…then" trips everyone up

In 1966 the psychologist Peter Wason introduced the selection task. Four cards show E, K, 4 and 7, and the rule says that if a card has a vowel on one side, it has an even number on the other. Which cards must you turn over to test the rule?

Most people choose E and 4. The rule only makes a promise about vowels. The 4 can have anything behind it: a consonant with an even number breaks nothing. The card that can break the rule is the 7, because a vowel behind it would be a vowel paired with an odd number. So the answer is E and 7.

The trap has a name: affirming the consequent, treating "if P then Q" as though it also meant "if Q then P". Once you spot it here, you'll see it in arguments everywhere.

Grid puzzles and the elimination table

The famous "Zebra puzzle" (five houses, five nationalities, five pets, who owns the zebra?) appeared in Life International in 1962. It's often credited to Einstein, but there's no evidence he wrote it. Puzzles like this are solved with a grid: rows for one category, columns for another, and a mark for every pairing a clue rules out.

Sudoku is a cousin. Howard Garns created it as "Number Place" for Dell Magazines in 1979, and the Japanese publisher Nikoli later gave it the name Sudoku. Every row, column and box must hold 1–9 exactly once, and every advanced technique is just a cleverer way of noticing which placements those three rules forbid.

A routine for when you're stuck

1. Rewrite every rule in your own words. Misreadings hide here.
2. Mark the clue you haven't used yet. It's probably the key.
3. Pick the cell or person with the fewest possibilities and split cases.
4. When a branch contradicts itself, write down why. That reason is often reusable.
5. Check the final answer against every rule, not just the last one.

Reasoning soundly matters more than reasoning quickly. A slow, airtight solution is worth more than a fast lucky one, because you can repeat it.

Sources

• Smullyan, R. What Is the Name of This Book? Prentice-Hall, 1978.
• Wason, P. C. "Reasoning", in New Horizons in Psychology, Penguin, 1966. Overview
• Zebra Puzzle history · Sudoku history