Lateral thinking, oriented

A puzzle that seems impossible usually rests on a detail you supplied yourself. You assumed the person is an adult, that a room has a window, that a number refers to people. Write down what you are picturing about who, where, when and what, then test each item against the exact words. The answer is usually ordinary once the wrong assumption is gone.

Read the question word for word. "Write nineteen without using any numbers" is satisfied by writing the word "nineteen", because a word is not a numeral. When you read quickly, your mind supplies the harder question it expected.

Check the premise before you calculate. Some questions ask about something that cannot exist, such as the change from a payment that was never made. If an answer seems to need arithmetic about an impossibility, the right option usually points out the impossibility instead.

Situation puzzles describe something strange and hide a plain cause: a game, a job, a performance, a photograph. Ask which everyday setting would make every detail normal at once, and reject explanations that fit most of the details but not all of them. If a solution turns into a long list of cases, stop and look for the single insight you have not used, such as a symmetry, an ordering or one object doing two jobs.

Rule: name your assumptions, read literally, and accept only the explanation that fits every detail.

Many trick questions are about what the sentence asks, not about arithmetic. "A shelf holds 12 books and you borrow 5. How many books do you have?" asks about your books, so the answer is 5. "All but 8 were sold" means 8 were not sold, so 8 are left, whatever the starting number. Paraphrasing takes a second and removes the pull of the numbers: "you are third in a queue and two people join behind you" becomes "nobody joined in front", so you are still third.

A quoted word in the question is a signal. Items like "what kind of 'press' has no buttons?" rule out one meaning and want another, so list the word's other lives. The press can mean news reporters, a press is a machine that squeezes or prints, and a press is also a lift in a gym. Keep the sense that fits every part of the clue.

Words also have technical meanings inside particular settings. A "menu" on a screen lists commands rather than dishes, and to "fold" egg whites in a kitchen is to mix them in gently. When a clue names a setting, start there.

Verbs are a frequent hiding place. You can break a habit, hold a grudge or draw a conclusion without touching anything. When a clue rules out physical handling, switch to the abstract sense of the verb.

Rule: paraphrase the question, find whose count it wants, and give every key word its other meanings.

Several classic puzzles offer a shortcut that is almost right. If a notebook and a pen cost $3.40 together and the notebook costs $3.00 more, the pen is not $0.40. Take the difference out first, $3.40 − $3.00 = $0.40, then halve it: $0.20. Check by adding back: $0.20 plus $3.20 is $3.40.

Doubling is easier backward. Dough that doubles every 10 minutes and fills its bowl at 4:00 was half full at 3:50, not at the halfway point of the time. A climbing puzzle needs its last day handled separately: a frog that climbs 7 feet a day and slips 2 each night in a 40-foot well is at 35 feet after 7 days and climbs out on day 8, with no slip.

Count gaps, not marks. A 60-meter fence with a post every 5 meters has 12 gaps and 13 posts. When everyone pairs with everyone once, multiply n by n − 1 and halve: 40 people make 780 handshakes. To sum a run, pair the ends: 11 + 12 + … + 30 is 10 pairs of 41, which is 410.

Clock angles need the hour hand's drift. The minute hand moves 6° a minute and the hour hand 0.5°. At 7:12 the hands sit at 72° and 216°, which is 144° apart. If the gap is over 180°, take 360° minus it for the smaller angle.

Rule: distrust the quick answer, and check it by rebuilding the story from your result.

Motion puzzles go wrong when two quantities are measured from different places. A passenger walking at 5 km/h on a train moving at 100 km/h moves at 105 km/h to someone on the platform and at 5 km/h to other passengers. Choose one observer first.

Average speed is total distance divided by total time, never the average of two speeds. Riding 30 km at 10 km/h takes 3 hours and riding back at 15 km/h takes 2, so 60 km in 5 hours is 12 km/h, not 12.5. More time is spent at the slower speed, which pulls the average down.

Some puzzles turn on what materials and light do. Light travels in straight lines, shadows fall away from the light, and a mirror swaps left and right in what it shows. Ice melts and heat fades, so evidence that has vanished may have changed state. If you can look only once, make a change now that leaves a trace you can read later, such as a warm surface or a moved marker. If behavior differs between two situations, find the one thing that differs.

Calendar and clock puzzles live at boundaries. A shop open from 9 p.m. Monday to 5 a.m. Tuesday is open for 8 hours across midnight, and a date a year from February 29 has no match in most years. Draw a short timeline and mark where the day, month or year turns over.

Rule: pick one observer, divide total distance by total time, and mark every boundary.

When a count of roles seems too high for the people present, someone is probably filling two roles. Two uncles and two nephews can be three people if the middle one is an uncle to the youngest and a nephew to the oldest. A parent is also a child, and a buyer may also be a seller.

Puzzles let you picture an average adult with average senses and an ordinary job. List what the story never says about the person: age, height, health, profession, whether they can see or hear. One of those unstated facts usually explains the scene. Relationship puzzles often turn on a gender or a generation the words never gave, so do not assume a doctor is a man or that a parent is a father. Label only what the sentence says.

Money puzzles invite you to add a sum that is already counted elsewhere. Follow one wallet or one till from start to finish, with every amount entering once and with its sign. If Jo borrows $10, repays $6 and then borrows $4 more, Jo owes $8.

Pricing puzzles hide the trap in a rule that covers only part of the purchase. Tickets at $12 each with $5 off the second ticket cost $12 + $7 = $19 for two, not $14. Apply each rule item by item.

Rule: look for double roles and unstated facts, and track money from one side only.

Some counting puzzles become easy once you sort by position. Cut a painted 4 × 4 × 4 cube into 64 small cubes. The 8 corners have three painted faces. Each of the 12 edges has 2 middle cubes with two painted faces, 24 in all. Each of the 6 faces has 4 center cubes with one painted face, another 24. The inner 2 × 2 × 2 block of 8 has none, and the four counts add back to 64, which is a useful check.

When a question says "of all sizes" or "in total", the larger shapes count too. A grid 5 cells long and 2 cells wide holds 10 squares of size 1 and 4 of size 2, 14 in all. Count by size, smallest first.

Chance puzzles reward listing over feeling. Two children with at least one girl gives the equally likely cases girl-girl, girl-boy and boy-girl, so the chance that both are girls is 1 in 3, not 1 in 2.

Estimation questions want the right order of magnitude, not the exact figure. Break the quantity into a few factors you can guess: hours slept in a lifetime is about 8 a night, 365 nights a year, for about 80 years, roughly 230,000 hours. Each factor may be a little off, but the product lands in the right range.

Rule: count by category, include every size, list the cases, and estimate with round factors.