How to solve short logic puzzles: constraints, elimination and checks

Turn the wording into constraints you can test

Identify the objects, their possible states and the relationships between them. For order, that means positions and words such as “before”, “after” and “immediately after”. For statements, establish which must be true and how many true statements are allowed. Do not import facts the puzzle never supplies.

“Before” does not mean “directly before”. “At least one” allows two or more; “exactly one” does not. “Not first” leaves every position except the first. Making these distinctions turns a vague story into conditions you can check one at a time.

You cannot simply reverse an “if” rule

Suppose the rule says: “If a card is gold, it has a stamp.” A gold card without a stamp breaks the rule. A stamped card, however, need not be gold: nothing forbids stamps on other cards. This distinguishes a conditional from its converse. [2]

The same rule implies that an unstamped card is not gold. Otherwise it would be a gold card missing its required stamp. Logic calls this the contrapositive. Both parts are negated and their order reversed; merely swapping the original parts does not work. [1]

OpenStax explains these argument forms and distinguishes valid conclusions from similar-looking mistakes. You do not need to memorise the terminology to play. Ask which specific case would break the rule.

An ordering puzzle: relationships before positions

Here is a separate teaching example. Four cards—a circle, star, diamond and square—stand in a row. The star is immediately after the circle. The diamond is before the circle. The square is after the star. Each card appears once. What is their order?

Combine the circle and star into a pair: circle → star. The diamond must precede that pair and the square must follow it. With four cards, the order is diamond → circle → star → square. Check each sentence: the star is the circle’s immediate right-hand neighbour, the diamond precedes the circle, and the square follows the star.

Now change only the last condition to “the square is not first”. Two arrangements work: diamond → circle → star → square, and diamond → square → circle → star. Both pass the checks. Finding one valid arrangement is not the same as proving it is the only one.

A truth puzzle: test a small set of cases

In another teaching example, a postage stamp is inside exactly one of three envelopes: A, B or C. A says “The stamp is in B”, B says “The stamp is not in B”, and C says “The stamp is not in A”. Exactly one inscription is true. Where is the stamp?

Do not trust the inscriptions at the outset. Assume each possible location in turn and count the inscriptions that would be true. A small case check replaces guessing which envelope is “lying”.

Check the answer and try to find another

Only A works in the envelope example: A’s inscription is false, B’s is true and C’s is false. If the question required “at least one true inscription”, all three locations would pass. The word “exactly” is what makes the solution unique.

After solving a puzzle, return to the original wording rather than your shortened notes. You may have dropped “immediately” or changed “not before” while paraphrasing. Then try to build a second valid result. If one exists, either you have more work to do or the question does not determine a unique answer.

A practical sequence for a one-minute puzzle

A one-minute format sets the pace of a game; it does not, by itself, measure intelligence. It is more useful to identify the condition you missed. You can approach the next puzzle with a more precise check, rather than simply demanding faster thinking.

1. Mark precise words: “exactly”, “at least”, “immediately” and negations.
2. Write down the objects and relationships without adding assumptions.
3. Use the tightest constraint or test the small set of possible cases.
4. Check the result against every condition and look for another valid answer.