This second collection calls for more than an instinctive answer. Some puzzles need a calculation; others require an invariant, a weighing plan or a strategic choice. Read the question, try to build a solution and only then reveal the result and explanation. When the original wording leaves an assumption implicit, we state it so that the answer can be checked.
The 50 puzzles
You have 8 balls that look identical, but one is heavier. What is the minimum number of weighings needed to find it with a balance scale?
Show result and explanation
Result: 2 weighings.
Explanation: Weigh 3 against 3. If they balance, compare the 2 excluded balls. Otherwise compare 2 from the heavier group of 3: if they balance, the third is heavier; otherwise the heavier of the two is the one.
You have 9 coins; one is counterfeit and lighter. How many weighings suffice?
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Result: 2 weighings.
Explanation: Split them into three groups of 3. Weigh two groups: the counterfeit is in the lighter group, or in the third if they balance. From the relevant group weigh one coin against another.
Water lilies cover a lake, doubling the area covered each day. If the lake is completely covered on day 48, when was it half covered?
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Result: Day 47.
Explanation: The final doubling changes half of the surface into the whole surface.
A father is four times as old as his son. In 20 years he will be twice as old. How old are they now?
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Result: Father 40, son 10.
Explanation: If the son is x years old, the father is 4x. In 20 years, 4x + 20 = 2(x + 20), giving x = 10.
Two trains are 300 km apart and travel towards each other at 80 and 70 km/h. How long until they meet?
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Result: 2 hours.
Explanation: Their closing speed is 80 + 70 = 150 km/h; 300 ÷ 150 = 2 hours.
A bat and a ball cost €1.10 together. The bat costs €1 more than the ball. How much is the ball?
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Result: €0.05.
Explanation: If the ball costs x, the bat costs x + 1: 2x + 1 = 1.10, hence x = 0.05.
A clock loses 5 minutes every hour. It is set correctly at noon. What will it show at 6 p.m. real time?
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Result: 5:30 p.m.
Explanation: Six real hours have passed and the clock has lost 6 × 5 = 30 minutes.
A man travels half the distance at 60 km/h and the other half at 40 km/h. What is his average speed?
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Result: 48 km/h.
Explanation: For two equal distances, the average is the harmonic mean: 2 ÷ (1/60 + 1/40) = 48, not 50.
Three switches control three light bulbs in a closed room. You may enter the room only once. How do you identify each switch?
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Result: Use light and heat.
Explanation: Turn on the first for a few minutes and switch it off; turn on the second and enter. The lit bulb belongs to the second switch, the unlit but warm one to the first, and the cold one to the third. This assumes bulbs that retain detectable heat; LEDs may not work.
You have two ropes, each of which burns in exactly 60 minutes but not at a uniform rate. How can you measure 45 minutes?
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Result: Light one rope at both ends and the other at one end.
Explanation: The first burns out in 30 minutes. Then light the other end of the second rope. It has 30 minutes of one-ended burning left and will now finish in 15 more minutes.
There are 100 closed doors. On pass 1 you toggle every door, on pass 2 every second door, on pass 3 every third door, and so on up to 100. Which remain open?
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Result: The perfect-square doors: 1, 4, 9, …, 100.
Explanation: Door n is toggled once per divisor of n. It remains open only if it is toggled an odd number of times.
Four people must cross a bridge at night. They take 1, 2, 5 and 10 minutes; they have one torch, the bridge holds at most two people, and a pair moves at the slower person’s speed. What is the minimum time?
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Result: 17 minutes.
Explanation: The 1- and 2-minute people cross (2); the 1-minute person returns (1); the 5- and 10-minute people cross (10); the 2-minute person returns (2); the 1- and 2-minute people cross (2). Total: 17.
A man must cross a river with a fox, a chicken and a bag of grain. The boat carries him and only one item. How does he do it?
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Result: Chicken, fox, chicken back, grain, chicken.
Explanation: Take the chicken across and return alone. Take the fox across and bring the chicken back. Take the grain across and return alone. Finally take the chicken across again. Fox and chicken, or chicken and grain, are never left alone together.
You have 12 coins; one is counterfeit, but you do not know whether it is heavier or lighter. What is the minimum number of balance-scale weighings?
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Result: 3 weighings.
Explanation: There are 24 hypotheses: any of 12 coins might be heavy or light. Two weighings have at most 3² = 9 outcome sequences; three have 3³ = 27. A three-weighing decision plan can distinguish all 24 hypotheses.
A car climbs a mountain at 30 km/h. How fast must it descend the same distance for the overall average to be 60 km/h?
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Result: No finite speed can do it.
Explanation: For an uphill distance d, it has already used d/30 hours, exactly the total time 2d/60 allowed by a 60 km/h round-trip average. There is no time left for the descent.
A bottle contains wine and water in a 3:2 ratio. If the total is 10 litres, how much wine is there?
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Result: 6 litres.
Explanation: There are 3 + 2 = 5 equal parts. Each is 10/5 = 2 litres; wine occupies 3 parts, or 6 litres.
A prize is hidden behind one of three doors. You choose one; a host who knows the prize location always opens one other empty door and offers you a switch. Should you switch?
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Result: Yes: switching wins with probability 2/3.
Explanation: Your first choice is wrong with probability 2/3. In that case the host must open the only other empty door, leaving the prize behind the door you can switch to. We assume the host always offers the switch.
You roll two fair six-sided dice. Is a total of 7 or a total of 8 more likely?
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Result: A total of 7.
Explanation: Six ordered outcomes sum to 7, while five sum to 8. Their probabilities are 6/36 and 5/36.
You have three boxes: apples only, oranges only and mixed. All labels are wrong. How many fruits must you draw to label them correctly?
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Result: One fruit.
Explanation: Draw from the box labelled “mixed”: it cannot be mixed, so the fruit reveals which single kind it contains. The two remaining labels then follow because both are also wrong.
You have an unmarked 5-litre jug and an unmarked 3-litre jug. You may fill, empty and pour between them until a jug is full. How can you measure exactly 4 litres?
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Result: Leave 4 litres in the 5-litre jug.
Explanation: Fill the 5-litre jug and pour into the 3-litre jug, leaving 2. Empty the 3-litre jug and transfer the 2 litres to it. Refill the 5-litre jug and pour 1 litre to fill the smaller jug; 4 litres remain.
Two people each play five games of chess, and each wins three. How is that possible?
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Result: They are not playing against each other.
Explanation: Each plays five games against other opponents, so both can win three.
A man lives on the 20th floor. Every morning he takes the lift to the ground floor. On returning he rides to the 12th and walks, except when it rains. Why?
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Result: He is short and uses his umbrella to reach the 20th-floor button.
Explanation: Usually he can reach only the 12th-floor button; on rainy days the umbrella lets him press the 20th.
You have 1,000 bottles, one poisoned, 10 mice and a poison that kills within 24 hours. How can you find the bottle in one testing round?
Show result and explanation
Result: Number the bottles in binary and let each mouse represent a bit.
Explanation: Number the bottles 0 to 999. Each mouse tastes a mixture from bottles whose number has a 1 in its assigned bit. After 24 hours, the affected mice form the poisoned bottle’s binary number, assuming reliable tests.
Four cards show A, D, 4 and 7. Every card has a letter on one side and a number on the other. Which cards must you turn over to test “if there is a vowel, the other side has an even number”?
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Result: A and 7.
Explanation: Turn A to check that its number is even. Turn 7 because its reverse must not be a vowel. D and 4 cannot disprove the rule.
The digits of a two-digit number add to 9. Reversing them increases the number by 27. What is it?
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Result: 36.
Explanation: If the digits are a and b, a + b = 9 and 9(b − a) = 27. Therefore b − a = 3, a = 3 and b = 6.
The digits of a two-digit number add to 11. Reversing them decreases the number by 27. Find it.
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Result: 74.
Explanation: If the digits are a and b, a + b = 11 and 9(a − b) = 27. Thus a − b = 3, a = 7 and b = 4.
What is the minimum number of people needed for the chance of a shared birthday to exceed 50%, assuming 365 equally likely, independent dates?
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Result: 23 people.
Explanation: Use the complement: the probability that all birthdays differ is (365/365) × (364/365) × … × (343/365). For 23 people it is about 0.493, so a match has probability about 0.507.
You have 7-minute and 11-minute hourglasses. How can you measure exactly 15 minutes?
Show result and explanation
Result: Start both; turn the 7-minute glass at minutes 7 and 11.
Explanation: At minute 7 the 7-minute glass empties: turn it. At minute 11 the 11-minute glass empties; turn the 7-minute glass again. It then has 4 minutes of sand in its upper chamber and empties at minute 15.
A farmer must divide 17 camels among three children: half to the first, a third to the second, a ninth to the third. How can he do it without cutting a camel?
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Result: Temporarily add an 18th camel.
Explanation: Compute the shares out of 18: 9, 6 and 2 camels. They add to 17, so the added camel can be taken back. The specified fractions do not add to 1; that is the trick of the story.
Can a chess knight visit every square of a 5×5 board exactly once and return to its starting square?
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Result: No.
Explanation: Each knight move changes square colour. A closed path must have an even number of moves; visiting all 25 squares once and returning would require 25 moves, an odd number.
Take an 8×8 chessboard and remove two opposite corners. Can you cover it with 31 dominoes of size 1×2?
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Result: No.
Explanation: The opposite corners have the same colour, leaving 30 squares of that colour and 32 of the other. Each domino always covers one square of each colour.
Choose five points in a square of side length 1. Can you guarantee that at least two are at most √2/2 apart?
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Result: Yes.
Explanation: Divide the square into four squares of side 1/2. By the pigeonhole principle, two of the five points lie in one small square; their distance is at most its diagonal, √2/2. Assign boundary points consistently to an adjacent square.
Three prisoners wear hats drawn from two white and three black hats. Each sees the other two hats but not their own. The first says “I don’t know”; the second says the same. The third knows their hat colour. What is it?
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Result: Black.
Explanation: Had the first seen two white hats, they would have known their own was black. The first answer rules that out. If the third wore white, the second could infer that their own was black; but the second also does not know. Thus the third wears black.
You have 25 horses and five racing lanes. You can race five at a time and have no stopwatch. What is the minimum number of races to identify the three fastest?
Show result and explanation
Result: 7 races.
Explanation: Run five group races and one race among the winners. If the winners finish A₁, B₁, C₁, D₁, E₁, A₁ is fastest. Only A₂, A₃, B₁, B₂ and C₁ can be second or third; a seventh race ranks them.
There are ten bags of coins. In nine bags every coin weighs 10 g; in one bag every coin weighs 9 g. How can one weighing on a digital scale identify the different bag?
Show result and explanation
Result: Take 1, 2, …, 10 coins from bags 1, 2, …, 10.
Explanation: The 55 coins would weigh 550 g if all were normal. If the actual weight is 550 − k grams, bag k has the 9 g coins. Assume each bag has at least ten coins.
You have two eggs and a 100-floor building. You want the highest floor from which an egg survives while minimising the worst-case number of drops. How many suffice?
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Result: 14 drops.
Explanation: Drop the first egg at floors separated by decreasing steps: 14, then 13, then 12 floors, and so on. If it breaks, use the second egg for a linear search in the last interval. The total is at most 14.
One of two doors leads to freedom. Two guards know which; one always tells the truth and the other always lies. You may ask one guard one question. What do you ask?
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Result: Ask which door the other guard would name, then choose the opposite door.
Explanation: The truthful guard reports the liar’s wrong answer. The liar lies about the truthful guard’s correct answer. Either way, the door named is wrong.
A man enters a bar and asks for a glass of water. The bartender points a gun at him. The man says thank you and leaves. Why?
Show result and explanation
Result: He had hiccups.
Explanation: The fright cures his hiccups, so he no longer needs water.
A woman pushes her car to a hotel and suddenly loses everything. Why?
Show result and explanation
Result: She is playing Monopoly.
Explanation: The car is her playing piece and the hotel is an expensive opponent’s property.
A man is found dead in a field with a closed backpack on his back. What happened?
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Result: His parachute failed to open.
Explanation: The “backpack” holds a parachute that stayed closed during the fall.
A man is found hanged in a locked room. There is only a puddle of water beneath him. How did he do it?
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Result: He stood on a block of ice.
Explanation: The block later melted, leaving the puddle.
Three people are in a room. Each shakes hands with every other person once. How many handshakes occur?
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Result: 3.
Explanation: There are 3 × 2 ÷ 2 = 3 pairs. We divide by 2 because each handshake involves two people.
If you raise a price by 20% and then reduce it by 20%, do you return to the starting price?
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Result: No: the result is 96% of the starting price.
Explanation: The combined multiplier is 1.20 × 0.80 = 0.96. The second 20% is applied to the already increased price.
A price falls by 50%. By what percentage must it rise to return to its original value?
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Result: 100%.
Explanation: After the reduction, half the price remains. Doubling that half restores the whole.
How many diagonals does a decagon have?
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Result: 35.
Explanation: Each vertex connects by diagonals to 10 − 3 = 7 vertices. Counting each diagonal twice gives 10 × 7 ÷ 2 = 35.
Complete the sequence: 2, 3, 5, 9, 17, ?
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Result: 33.
Explanation: The proposed rule is “double and subtract 1”: 17 × 2 − 1 = 33. As with any finite sequence, other continuations are possible without an explicit rule.
Find the missing number: 1, 2, 6, 24, 120, ?
Show result and explanation
Result: 720.
Explanation: These are factorials from 1! to 6!, and 6! = 720. Other rules could reproduce the first terms, but this is the natural one.
What comes next: 1, 4, 10, 22, 46, ?
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Result: 94.
Explanation: The proposed rule is “double and add 2”: 46 × 2 + 2 = 94. A finite sequence alone does not determine a unique next term.
A man must move 3,000 bananas over 1,000 km. A camel can carry at most 1,000 and eats one banana per kilometre travelled. What is the maximum delivery, allowing intermediate caches and fractional bananas?
Show result and explanation
Result: 533⅓ bananas in the continuous model.
Explanation: Moving 3,000 bananas until 2,000 remain costs five camel-kilometres per kilometre of progress: 1,000/5 = 200 km. Moving 2,000 to 1,000 costs three per kilometre: 1,000/3 = 333⅓ km. The remaining distance is 466⅔ km with one load of 1,000, so 1,000 − 466⅔ = 533⅓ arrive. Indivisible bananas require a separate discrete model.
Three people pay €30 for a room. The hotel returns €5. The porter keeps €2 and gives €1 back to each guest. Each paid €9, totalling €27; adding the porter’s €2 gives €29. Where is the missing euro?
Show result and explanation
Result: No euro is missing.
Explanation: The €27 already include the porter’s €2: €25 went to the hotel and €2 to the porter. The €3 returned to guests complete the original €30. Adding €2 to €27 again is the mistake.