Logic puzzles, oriented

Restate every clue in a compact form before you reason with it: an arrow for a comparison, "not" for an exclusion, a number for a fixed position. Written constraints can be checked against each other, while clues held in your head blur. Most wrong answers come from a clue remembered slightly wrong.

Start with the piece the clues pin down most tightly, settle it, and remove that option from everyone else. Each settled piece usually unlocks the next. Elimination counts as proof: when only one option survives every clue, it is the answer as surely as if a clue named it.

In an each-gets-a-different-one puzzle, negative clues do most of the work. Rosa, Theo and Una each play a different sport: tennis, golf or swimming. Rosa plays neither tennis nor golf, so she swims. Theo does not play golf, and swimming is taken, so he plays tennis. Una, who appears in no clue, plays golf. Solvers who wait for a clue about Una choose "cannot tell", but the empty slot is hers.

For larger puzzles, draw a grid with people down the side and options across the top, and mark each yes and no. For a shared choice, such as a meeting day, intersect everyone's options: every constraint applies at once.

When elimination stalls, list the two or three arrangements still possible and test each against every clue. The one that survives is the answer.

Rule: write the clues down, settle the tightest piece first, and trust elimination.

Turn comparisons into one chain before answering. "Rosa is older than Theo" and "Theo is older than Una" give Rosa > Theo > Una, so the youngest is the bottom of the chain. A clause like "Theo is taller than Una but shorter than Rosa" puts Theo in the middle, even though he is named first.

Place fixed positions first. "First", "last" and "at position 2" go in at once, and "not last" or "not next to" become decisive only once few slots remain. "Immediately to the right" means exactly one step from a known point.

Rows have two counting traps. The person in the exact middle of a row of 15 has (15 − 1) ÷ 2 = 7 people on each side, not 7.5. Someone 12th from the front and 9th from the back is counted in both numbers, so the line holds 12 + 9 − 1 = 20 people. At a round table, everyone faces the center, so judge left and right from the seated person's view.

Day riddles move one day per time word. If the day after tomorrow is Monday, tomorrow is Sunday, today is Saturday and yesterday was Friday. For long jumps, use the remainder after dividing by 7: 100 days after a Tuesday is a Thursday. For a walking route, total the north-south legs and the east-west legs separately; opposite legs cancel, and the totals name the direction.

Rule: build one ordered line, then count every step and every end exactly once.

When two groups overlap, adding them counts the overlap twice. If 22 people play chess, 17 play go and 6 play both, the group holds 22 + 17 − 6 = 33 people.

"At least" and "at most" questions ask about the overlap itself. In a class of 40, if 31 wear glasses and 28 ride the bus, there are 59 memberships among 40 students, so at least 59 − 40 = 19 must be in both. The overlap can be as large as the smaller group, 28, if every bus rider wears glasses. Read which one the question asks for.

Things that can be shared are counted once. If each of 4 brothers has 2 sisters, the sisters are the same two girls for every brother, so the family has 6 children, not 12.

Family riddles are solved from the inside out. Replace each possessive phrase with the person it names, then reread the sentence. "My mother's mother's only daughter" is my mother, because "only" rules out anyone else. When a man says "my father's only son", he means himself. Errors nearly always come from an assumption the words never gave: that a parent is a father, or that "grand" is one generation. Sketching the people as boxes, parents above children, makes those assumptions visible.

Rule: subtract an overlap once, count shared people once, and resolve every phrase to a name before you answer.

A conditional rule makes one promise: whenever P happens, Q happens. The only way to break it is a case with P and no Q. So when you test the rule, check anything showing P, to see whether Q is with it, and anything showing not-Q, to see whether P is hiding there. Cases showing not-P or Q cannot break the rule and can be skipped. For "anyone drinking alcohol must be over 21", check the beer drinker and the 16-year-old, not the cola drinker or the 25-year-old.

One backward step is valid. From "if the oven is on, the light is lit" and "the light is off", it follows that the oven is off. This is called modus tollens, or denying the consequent.

The tempting backward step is invalid. From the same rule and "the light is lit", nothing follows about the oven, because something else may have lit the light. Treating a rule as if it ran both ways is called affirming the consequent, and it is the trap these items are built around.

Universal rules work the same way. "Every square is a rectangle" lets you go from square to rectangle, never from rectangle to square. Before choosing an answer, ask whether it follows the arrow forward or quietly reverses it.

Rule: a conditional can be checked forward from P or backward from not-Q, and never backward from Q.

Puzzles with truth-tellers and liars are solved by testing cases, not by intuition. Assume one person tells the truth, work out what that makes everyone else, and check every statement for consistency. If you reach a contradiction, the assumption was wrong, and its opposite is now proved.

Suppose Kim says "Lee lies", Lee says "Max lies" and Max says "Kim and Lee both lie". If Kim tells the truth, Lee lies, so Max tells the truth, but Max says Kim lies, which contradicts the assumption. So Kim lies. Then Lee tells the truth and Max lies, and Max's claim is indeed false.

On an island of knights, who always tell the truth, and knaves, who always lie, ask which type could make each statement. If A says "we are both knaves", a knight could not say it, because it would be false. So A is a knave, the statement is false, and B must be a knight. Some statements cannot be made by either type at all, which is itself a clue.

When an item says that exactly one statement is true, or exactly one person lies, try each candidate in turn as the true one. Usually only one choice leaves everything consistent.

The common error is following a single statement without checking the rest. A case is only alive if every statement agrees with it.

Rule: assume, follow every consequence, and keep only the case with no contradiction.

A two-pan balance can tip left, tip right or balance, so one weighing distinguishes three possibilities. Split the coins into three equal groups and weigh two: the result names the group holding the heavy coin. Each weighing divides the candidates by 3, so 4 weighings handle 81 coins. Splitting the coins in half wastes the balanced outcome.

Some puzzles yield to counting what each step changes for good. In a knockout tournament with 50 players, every match removes exactly one player and 49 must go, so there are 49 matches however the bracket is drawn.

Number-clue grids hide items under covered squares, and each number counts the items in the squares touching it, diagonals included. Start with certainties. A number with exactly as many covered neighbors as its value means all of them hold an item. A number whose count is already met means its other covered neighbors are empty. Work in passes: mark what is certain, then revisit every number beside a mark.

When no single number settles anything, compare neighbors whose squares overlap. If a 1 touches two covered squares and the 2 beside it touches those two plus one more, the 1 accounts for one item among the shared pair, so the extra square must hold the 2's other item.

Rule: look for the step that removes the most possibilities at once, and count that instead of simulating.