Logic Puzzles curriculum

Restate each clue in a compact form before using it: "Rosa > Theo" for a comparison, "Theo ≠ golf" for an exclusion, "bakery = 1" for a placement. Written constraints can be checked against each other, while clues held in your head blur together. Most errors in these puzzles come from a clue remembered slightly wrong, not from faulty reasoning.

Look for the person, slot or item that the clues pin down most tightly, and settle it first. Each settled piece removes an option from everyone else, so the rest of the puzzle usually falls into place in a chain. Starting with a loosely constrained piece leads to guessing and backtracking.

List all the options, then cross off whatever a clue rules out. When only one option is left for a slot, you have proved it rather than guessed it. This works even when no clue mentions the answer directly. The survivor of elimination is the answer as surely as one named outright.

Puzzles often leave one person out of the clues on purpose. Solvers then wait for a clue about them, see none, and choose "cannot tell". But once everyone else is placed, the one remaining slot belongs to the unmentioned person. Absence of a clue about someone is a clue in itself.

Some clues leave two or three arrangements open, such as "directly above" in a small building. Write each arrangement out, then check it against every other clue. A case that breaks any clue is gone, and the case that survives is the answer. Contradiction is a tool, not a dead end.

Write each comparison as an arrow and join them: "Rosa is older than Theo" and "Theo is older than Una" give Rosa > Theo > Una. Only then answer: the youngest is the bottom of the merged line. The trap is answering from one clue, often by picking the first person named, without building the whole chain.

A clause like "Theo is taller than Una but shorter than Rosa" gives Rosa > Theo > Una, so Theo sits between the other two. Because Theo is named first, solvers often place him at an end. Rewrite the clause as two arrows that both touch Theo, and his middle position becomes obvious.

Clues like "first", "last" or "at position 2" go in immediately. Clues like "not last" or "not next to" only become decisive once the open slots are few. In a five-person line with first and second already filled, "not last" can settle two people at once. Filling the fixed slots first is what gives the negative clues their power.

Anchor on the item whose position you know, then move exactly one place. In a row numbered from left to right, immediately right of position 3 is position 4 and immediately left is position 2. The common slips are moving the wrong way or moving two places, so say the direction aloud as you step.

Take away the middle person and split the rest evenly. In a row of 15, the middle person has (15 − 1) ÷ 2 = 7 people on each side. The tempting answer is half the row, 7.5 or 8, which forgets that the middle person stands on neither side.

If someone is 12th from the front and 9th from the back, the front count includes them and so does the back count. Adding gives 21, but that counts them twice, so the line holds 12 + 9 − 1 = 20 people. Picture a line of three where the middle person is 2nd from both ends to see the one you subtract.

Everyone at a round table faces the center, so a person's right is the seat to their right as they sit, which is your left when you look at them across the table. Seat the fixed pair first, such as two people opposite each other, then use the left and right clues from the seated person's view to place the rest.

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. That leaves golf for Una, who appears in no clue at all. Settle the person with the most exclusions first and let the rest cascade.

When there are two attributes to match, such as pets and shirt colors, draw a table with people down the side and options across the top. Mark an X for every exclusion and a check for every certainty. Each check lets you X out the rest of its row and column, which often creates a new certainty somewhere else.

For a shared meeting day, write the full week, then strike out each day any person cannot make. All constraints apply at once, not one after another. If one person can only do Monday, Wednesday or Friday, another cannot do Wednesday and a third cannot do Monday, only Friday survives.

Pin the named day, then hop. If the day after tomorrow is Monday, then tomorrow is Sunday, today is Saturday and yesterday was Friday. The usual error is compressing two time words into one hop, so count every word on your fingers.

To find the day 100 days after a Tuesday, divide 100 by 7: the remainder is 2, so it is two days after Tuesday, a Thursday. Whole weeks bring you back to the same weekday, so only the remainder moves you. For days before a date, count the remainder backward.

Add up the north and south legs as one total and the east and west legs as another. A walk of 6 blocks west, 3 north, 6 east and 8 south nets to 0 east-west and 5 south, so the walker ends due south of the start. The shape of the path does not matter, only the totals.

If 22 people play chess, 17 play go and 6 play both, adding 22 and 17 counts those 6 twice. So the group holds 22 + 17 − 6 = 33 people. Whenever two groups overlap, subtract the overlap once to count each person exactly once.

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. Keep the groups apart for as long as you can; the memberships that do not fit inside the total have to be shared.

Overlap is largest when one group sits entirely inside the other. In the class of 40 with 31 wearing glasses and 28 riding the bus, at most 28 can do both, because the bus group is only 28 strong. "At least" and "at most" questions have different answers, so check which one is asked.

If each of 4 brothers has 2 sisters, the sisters are the same two girls for every brother. The family has 4 boys and 2 girls, 6 children, not 4 + 8. Siblings, neighbors and handshakes are shared between people, so draw the group once instead of multiplying.

A rule "if P then Q" is broken only by a P with no Q. So check anything showing P, to see whether Q is with it, and anything showing not-Q, to see whether P is hiding there. For the rule "anyone drinking alcohol must be over 21", check the beer drinker and the 16-year-old, and skip the cola drinker and the 25-year-old, who cannot break it.

From "if the oven is on, the light is lit" and "the light is off", it follows that the oven is off. This step, called modus tollens, is valid because an oven that was on would have lit the light. It is the conclusion these items most often ask for.

From "if the oven is on, the light is lit" and "the light is lit", you cannot conclude the oven is on, because something else may have lit it. Treating the rule as if it ran both ways is called affirming the consequent. Confirming the "then" half proves nothing about the "if" half.

"Every square is a rectangle" tells you a square has four right angles, but not that a rectangle is a square. Read a universal rule as a one-way arrow from membership to property. Before choosing an answer, ask whether it uses the arrow forward or quietly reverses it.

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.

Suppose A says "we are both knaves". A knight could not say that, because it would be a lie. So A is a knave and the statement is false, which means they are not both knaves: B is a knight. For each speaker, ask which type could make that statement without breaking the rules.

Items that say "exactly one of these is true" or "exactly one person lies" are solved by trying each candidate. Make one statement true and all the others false, then check for a contradiction. Usually only one choice leaves everything consistent, and that choice is the answer.

Work from the inside out. "My mother's mother's only daughter" is my mother, because my mother is a daughter of her own mother and "only" rules out anyone else. Replace each resolved phrase with a plain name, such as "my mother", and the riddle usually becomes one short sentence you can read directly.

A son of my father is me or my brother, and "only" removes the brother. So when a man says "my father's only son", he means himself. Watch for "only" in these riddles; it almost always points back to the speaker and closes off a sibling the wrong answers depend on.

Sketch the people as boxes, with parents above children, and add each relationship as you read it. Errors in these riddles nearly always come from an assumption the words never gave: that a parent is a father, or that "grand" is one generation when it is two. A drawing makes those assumptions visible.

A two-pan balance can tip left, tip right or balance. Split the coins into three equal groups and weigh two of them: the result names the group with the heavy coin. Each weighing divides the candidates by 3, so w weighings handle up to 3^w coins. For 81 coins that is 4 weighings, since 3^4 = 81. Halving the coins wastes the balanced outcome.

In a knockout tournament with 50 players, every match eliminates exactly one player, and 49 must be eliminated to leave a champion. So there are 49 matches, whatever the bracket looks like. Many hard puzzles fall to one reframing like this, rather than to brute force.

A number tells you how many of its neighboring covered squares, including diagonal ones, hold a hidden item. A 0 means none of them do, so all its covered neighbors are safe. Read each number as a small equation about the squares around it, and use the edges of the grid, where a square has fewer neighbors.

If a 2 touches exactly two covered squares, both must hold an item. Scan the grid for these saturated numbers first, because they give certain answers with no reasoning beyond counting. Mark those squares, then move to the numbers next to them.

Once a number has as many marked items around it as its value, any covered squares left around it must be empty. Work in passes: mark the certain items, then revisit every number beside a mark and subtract. Each pass usually unlocks the next.

If one number's covered squares all sit inside a neighbor's, subtract. Suppose 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 2's extra square must hold the other. This subset step solves boards where every single number looks open.