Back to 162 Logic Puzzles & Brain Teasers With Answers

162 Logic Puzzles & Brain Teasers With Answers

162 riddles · answers on the last page · riddleup.com/riddles/logic-puzzles-and-brain-teasers

  1. A doctor gives you three pills and tells you to take one every half an hour. How long will the pills last?
  2. Some adults still use these to count.
  3. Sophie's mother had 2 sons and 1 daughter. The sons names were George and John. What was the daughter's name?
  4. I run around the city, but I never move.
  5. You are in a room with 3 monkeys. One monkey has a banana, one has a stick, and one has nothing. Who is the smartest primate?
  6. Where is the ocean deepest?
  7. What's at the head of an elephant and at the tail of a squirrel?
  8. 2 people in front of 2 people. 2 people behind 2 people, and 2 people beside 2 people. How many people are there?
  9. What word is that, which, deprived of its first letter, leaves you sick?
  10. Besides Paris, what is the capital of France?
  11. What part of a fish weighs most?
  12. What was was, before was was was?
  13. I you were to take two apples from three apples, how many would you have?
  14. What is round on both ends and hi in the middle?
  15. Where can you add 2 to 11 and get 1?
  16. If it has a quart capacity, how many pennies can you put into a empty piggy bank?
  17. What is between heaven and earth?
  18. A farmer has 17 sheep and all but 9 run away. How many are left?
  19. How can the letters OWONDER be rearranged to make one word?
  20. John's mom had three children. The first child was April. The second child was May. What was the third Child's name?
  21. What is the largest living ant on earth?
  22. What has three feet but no arms or legs?
  23. What follows a dog wherever it goes?
  24. What is never eaten before lunch?
  25. I met a man and drew his name. He tipped his hat and played my game. What is his name?
  26. If you were standing directly on Antarctica's South Pole facing north, which direction would you travel if you took one step backward?
  27. Feed me and I live, yet give me a drink and I die. What am I?
  28. What's higher than the king?
  29. Look at me one way and I weigh a whole lot; turn me around and you'll see that I am not. What am I?
  30. I turn around once. What is out will not get in. I turn around again. What is in will not get out. What am I?
  31. The sun bakes them, The hand breaks them, The foot treads them, The mouth tastes them.
  32. What breaks on the water but never on the land?
  33. What can you keep after giving it to someone?
  34. There is a boat with a ladder attached to it, the ladder is eight feet tall. If the water rises four feet how much of the ladder will be on top of the water?
  35. Susan and Lisa decided to play tennis against each other. They bet $1 on each game they played. Susan won three bets and Lisa won $5. How many games did they play?
  36. What is so delicate that even mentioning it breaks it?
  37. There is a clerk at the butcher shop, he is five feet ten inches tall, and he wears size 13 sneakers. What does he weigh?
  38. What would you father's only sister's sister-in-law be to you?
  39. Timmy's mother has three children. The first was named April. The next was named May. What is the final one's name?
  40. George, Helen, and Steve are drinking coffee. Bert, Karen, and Dave are drinking soda. Is Elizabeth drinking coffee or soda?
  41. MT_TF_S What are the missing letters? Why?
  42. A mother had five boys Marco, Tucker, Webster and Thomas. Was the fifth boys name Frank, Evan or Alex?
  43. What gets broken without being hold?
  44. There is a word in the English language in which the first two letters signify a male, the first three letters signify a female, the first four signify a great man, and the whole word, a great woman. What is the word?
  45. The more you take from me, the bigger I get. What am I?
  46. What's always coming, but never arrives?
  47. If a rooster laid a white and brown egg, what kind of chicks would hatch?
  48. Homeless man bets he can sing a song about "Frederica Armstrong Miller."
  49. Payoff 2n (St. Petersburg). Charge to play?
  50. Rule: Circle=Yellow. Cards: C, S, Y, R. Check?
  51. Probability to be the last to try pitcher?
  52. Surprise test impossible logic?
  53. Sphere or Torus using rocks?
  54. 9 correct predictions in a row. Psychic?
  55. Describe a chicken in computer programming.
  56. Hourglass weigh more falling or at rest?
  57. Pays payout in Yen. Charge to play?
  58. 5 black spins. Is red "due"?
  59. 10 pennies in 3 cups, each has odd number.
  60. Spoon tea into coffee, mix, spoon back. More what?
  61. Max slices from 4 straight cuts?
  62. Ways to arrange 8 books on a shelf?
  63. 1 poisoned bottle in 1000. Prisoners needed?
  64. P1 (4 of kind), P2 (3 of kind). Worst possible hand?
  65. Break stick at 2 random points. Triangle prob?
  66. Monkey climbs rope with matching weight. Weight?
  67. Reverses left/right but not up/down?
  68. Camel 3000 bananas, 1k km. Max reach?
  69. Get fair toss from biased coin.
  70. Spheres with 6cm holes. Which has more volume?
  71. 200 fish, 99% Gold. Remove how many for 98%?
  72. Brothers Chris (FL) and Connor (OR) same time. How?
  73. Irrational number ^ irrational power = rational?
  74. 1st sits random. Last gets assigned seat prob?
  75. Does washer hole expand or shrink when heated?
  76. How many barbershops are in the USA?
  77. 100 cheating husbands. Execution day?
  78. Truth 3/4 times. Says 6 on a die. Prob it's a 6?
  79. 5 bandits split 100 coins. Senior proposal?
  80. Families stop childbearing after 1st boy. Sex ratio?
  81. How many squares on an 8x8 chessboard?
  82. How many ping pong balls fill a 747?
  83. C beats T by 10m in 100m. C starts 10m back. Winner?
  84. Does level change when ice in water melts?
  85. 4 people: 1, 2, 5, 10 mins. Bridge (max 2). Min time?
  86. 31 games in elimination tournament. Players?
  87. 3 doors: Heaven, 1-day Hell, 2-day Hell. Avg days?
  88. 110m rope. 1ft slack added/pulled up. Truck fits?
  89. Which is greater: e? or ?e?
  90. Surgeon with 2 gloves must operate on 3 patients. How?
  91. Duck in pond. Cat runs 4x faster on edge. Escape?
  92. 1/2 of 2/3 of 3/4... of 9/10 of 1,000?
  93. Pick 1 of 3. Empty removed. Do you switch?
  94. Explain a database to an 8-year-old.
  95. Standard deviation of {1, 2, 3, 4, 5}?
  96. 1/4 into tunnel. Meet train at start or end. Speed ratio?
  97. Sum of digits from 1 to 1,000,000?
  98. Helium balloon in car accelerating forward. Direction?
  99. HH = Jamie takes 2. Any other = Ashley takes 1. Winner?
  100. 100 ants on 1m stick at 1cm/s. Max time to clear?
  101. Shortest distance from room corner to opposite corner?
  102. Measure 4L using 3L and 5L jugs.
  103. Angle between clock hands at 3:15?
  104. 3-way. Accuracy 33/50/100. You (33) go first. Action?
  105. 20% men/40% women attend. 35% employees men. Total %?
  106. How is world population measured?
  107. Alex (RR/BB wins). Sam (BR/RB wins). Better position?
  108. 7-part bar. Pay 1 part/day for 7 days. Snap twice?
  109. Weigh an elephant without a massive scale?
  110. 3R, 2B hats. 1st/2nd can't guess. What is 3rd's color?
  111. How do you move Mount Fuji?
  112. Double every 5m. 1 fills in 12h. Time for 2 to fill?
  113. Guess 1-1000 with higher/lower. Max guesses?
  114. Can the mean of two consecutive primes be a prime?
  115. Can you swim faster in syrup or water?
  116. 6 chambers, 2 consecutive bullets. Fired empty. Spin or stay?
  117. Times digit "3" is printed between 3,000 and 4,000?
  118. 1) 1 penalty for �1k. 2) 2 of 3 for �1k. Choice?
  119. Place coins on round table. Last one wins. Go 1st or 2nd?
  120. 54 cards (2 Jokers). Probability both are Jokers?
  121. Name 5 uses for a stapler without staples.
  122. 14ft pole. Up 5ft day, down 3ft night. Days to top?
  123. Owe 49+49=98. Plus your $1 = 99. Where is the dollar?
  124. 1/1000 trick coin. Flip 10 heads. Prob it's the trick coin?
  125. How can the Earth's mass be measured?
  126. Pick 1 of 1000 seats. 990 empty removed. Odds now?
  127. Price reduced 23%. Increase needed to restore original?
  128. 100B and 100W pearls. Maximize chance of picking Black.
  129. How many smartphones are there in London?
  130. Two 1-hr ropes, burn inconsistently. Measure 45 mins?
  131. Black win $1, Red lose $1. Quit anytime. Strategy?
  132. Does running keep you drier than walking in rain?
  133. Ascend 9-7 Day 1, descend 9-7 Day 2. Spot reached at same time?
  134. 25 horses, 5-horse races. Min races to find top 3?
  135. Flip 4 coins. At least 2 are tails. Probability exactly 3 are tails?
  136. 10R, 20B, 30G socks in 3 sizes. Min pull for matching pair?
  137. How many corners (vertices) does a tesseract have?
  138. Joey/Charlie share 50ml. Joey sips 1/2, Charlie 1/2 of rest, repeat...
  139. Why are manhole covers round?
  140. Measure 9 mins using 4-min and 7-min hourglasses.
  141. Choice: 1) Roll 1 die for $1M x result. 2) Roll 1M times for $1 x total.
  142. Brick thrown from boat into small pond. Does water level change?
  143. How many times in 24 hours do clock hands form a 90� angle?
  144. 20-ton lorry on 20-ton bridge. Bird lands halfway. Does it collapse?
  145. Family has 2 kids, one is a boy. Probability the other is a girl?
  146. 5 cats catch 5 mice in 5 mins. Cats needed for 100 mice in 100 mins?
  147. 100 people flip switches of multiples of their entry. How many stay on?
  148. How many gas stations are there in the United States?
  149. 2 A and 2 B pills identical. Hand has 1A, 2B. How to get correct dose?
  150. 10x10x10 cube painted then cut into 1x1x1s. How many have no paint?
  151. 3-pointer (40% win) or 2-pointer (70% tie, then 50% OT win)?
  152. 9 marbles, one is heavier. Min weighings on a balance to find it?
  153. What is the sum of all the integers from 1 to 100?
  154. You measure my life in hours and I serve you by expiring. I'm quick when I'm thin and slow when I'm fat.
  155. I am tall when I'm young and short when I'm old. I can brighten your day unless it's windy.
  156. I have lakes with no water, mountains with no stone and cities with no buildings.
  157. If you drop me I'm sure to crack, but give me a smile and I'll always smile back.
  158. I am always hungry and will die if not fed, but whatever I touch will soon turn red.
  159. I turn once, what is out will not get in. I turn again, what is in will not get out.
  160. I have lakes with no water, mountains with no stone and cities with no buildings. What am I?
  161. What has words, but never speaks?
  162. What is cut on a table, but is never eaten?

Answers

  1. One hour.You take the first pill immediately, the second after 30 minutes, and the third after another 30 minutes, a total span of one hour, not three half-hours.
  2. FingersMany adults still count on their fingers, even though it's often thought of as a childhood habit.
  3. SophieThe riddle is about 'Sophie's mother,' so Sophie herself must be that daughter.
  4. A wallA city wall surrounds the whole city while staying completely stationary.
  5. You, since you are the only human, and humans are the smartest primates.Humans are primates too, and the riddle's trick is realizing the reader, standing in the room, outranks all three monkeys in intelligence.
  6. At the bottom.This is a joke answer: no matter where you measure, the ocean is always deepest at its very bottom.
  7. The letters "el"Both 'ELephant' and 'squirr-EL' contain the letters 'EL' -- at the start of one word and the end of the other.
  8. FourFour people standing in a square arrangement satisfy all three conditions at once, since each pair is simultaneously in front of, behind, and beside another pair.
  9. MusicThis relies on an old, loose pun: removing the first letter from 'MUSIC' leaves 'USIC,' which said aloud sounds a bit like 'you sick' (as in 'are you sick?'). It's a stretch of wordplay rather than an exact letter-for-letter fit.
  10. The letter "F"This is a play on words: the 'capital' (first, capitalized letter) of the word 'France' is F.
  11. The scalesThis is a pun: a fish's 'scales' are what you'd use to weigh it, since 'scales' also means a weighing instrument.
  12. IsThis wordplay riddle points out that the past-tense word "was" used to be the present-tense word "is" before events became past.
  13. 2You physically took 2 apples, so you now have 2 in your hand, regardless of how many remain elsewhere.
  14. OhioSpelled out, "Ohio" is round letters (O) on both ends with "hi" sitting right in the middle.
  15. On a clock. 11 o'clock plus 2 hours equals 1 o'clock.On a 12-hour clock face, 11 o'clock plus 2 hours wraps around back to 1 o'clock.
  16. One, after that it is no longer empty.This is a trick riddle, not a math problem: you can put in only one penny, because after that the piggy bank is no longer "empty."
  17. The word "and"In the phrase 'heaven AND earth,' the word 'and' is literally positioned between the two words.
  18. Nine"All but 9" means 9 sheep stayed behind and did not run away, so nine remain.
  19. "One word" — the letters spell out the phrase 'one word' itself.OWONDER rearranges, letter for letter, into the phrase "ONE WORD" -- the answer is also the method for solving it.
  20. JohnThe riddle names 'John's mom,' so naturally the third child being asked about is John himself.
  21. An elephantHidden inside the word 'elephant' is the word 'ant,' and elephants are famously enormous.
  22. A yardA yard is a unit of length equal to exactly three feet, though it has no literal arms or legs.
  23. Its tailA dog's tail is physically attached to it, so it trails behind the dog no matter where it goes.
  24. DinnerDinner is the meal that comes after lunch, so by definition it is never eaten before it -- a simple sequence riddle rather than wordplay.
  25. AndrewThis is a pun on 'and drew,' hidden inside the sentence itself, spelling the name 'Andrew.'
  26. NorthAt the South Pole, every direction you face or move is technically north, since it's the southernmost point on Earth -- so even a "backward" step still takes you north.
  27. FireFeeding fire with fuel keeps it burning ('alive'), but pouring water on it ('a drink') extinguishes it -- a rephrased version of an earlier fire riddle.
  28. A crownA king's crown physically sits above his head, making it 'higher' than the king himself.
  29. A tonThe word "ton" (a heavy unit of weight) spelled backward is "not," meaning something that isn't heavy at all -- a clever visual/word trick.
  30. A keyTurning a key one way locks a door (keeping intruders out); turning it back the other way unlocks it (letting whoever's inside back out).
  31. GrapesGrapes ripen in the sun, are picked by hand, were traditionally crushed underfoot in winemaking, and are also eaten and tasted directly.
  32. A waveOcean waves crest and "break" on the water's surface, but this specific kind of breaking doesn't occur on dry land.
  33. Your word.Giving someone 'your word' (a promise) doesn't mean you lose it -- you still keep your word by honoring the commitment.
  34. Eight feet, the ladder is attached to the boat.Since the ladder is attached to the floating boat, it rises with the water level, so the entire eight-foot ladder stays above the water regardless of the tide.
  35. They played eleven games.If Susan won 3 games and Lisa's net winnings were $5, Lisa must have won 8 games (8 - 3 = $5 net), so together they played 3 + 8 = 11 games.
  36. SilenceThe word 'silence' is broken the instant you speak it out loud -- saying it defeats the very thing it names.
  37. MeatA butcher shop clerk weighs meat for customers, so the physical details about the clerk himself are just a distraction.
  38. Your mother.Working through the family relations, a father's only sister's sister-in-law circles back to being your own mother.
  39. TimmyThe riddle names 'Timmy's mother,' so the third child sharing that family is simply Timmy himself.
  40. Coffee. The letter E appears twice in her name, as it does in the names of the others drinking coffee.George, Helen, and Steve (the coffee drinkers) each have the letter "e" appearing twice in their names, matching Elizabeth's name, which also contains two "e"s -- so she is drinking coffee too.
  41. W and S, for the days of the week (Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday).The letters given (M, T, _, T, F, _, S) are the first letters of the seven days of the week, with Wednesday's W and Sunday's S missing from the blanks.
  42. Frank. Each child's first name begins with the first letters of the days of the week: M, T, W, T, Frank.Look at just the first letters of each brother's name to spot the days-of-the-week pattern.
  43. A promiseA promise is an intangible commitment; it can be "broken" (not honored) despite never being a physical object one holds.
  44. Heroine"He" signifies a male, "Her" signifies a female, "Hero" signifies a great man, and the full word "Heroine" signifies a great woman -- each stage of the word builds in meaning exactly as described.
  45. A holeDigging out (removing material from) a hole only makes the hole itself larger.
  46. Tomorrow"Tomorrow" is always one day ahead of today -- once it comes, it has become "today," so tomorrow itself never truly arrives.
  47. NoneThink about the literal properties of a 'nan'. The riddle uses metaphors to describe its physical form or function.
  48. "happy birthday."Think about the literal properties of a '"happy birthday."'. The riddle uses metaphors to describe its physical form or function.
  49. Small amountThe mathematical expectation is infinite, but most people wouldn't pay more than �20. This illustrates that humans value money based on 'utility' rather than just the raw expected amount.
  50. Circle and RedYou check the Circle (to see if back is Yellow) and the Red card (to see if front is a Circle, which would break the rule). Yellow and Square can't invalidate the rule.
  51. 1/n (Equal)This is a property of a 'random walk' on a circle. Every person is equally likely to be the last one reached before the circle is completed.
  52. ParadoxIf the test hasn't happened by Thursday, the students know it must be Friday (so it's not a surprise). This logic continues backward to Monday, 'proving' it can never be a surprise.
  53. Perpendicular loopsOn a sphere, any two 'great circles' must intersect. On a torus, you can have a loop that goes around the center and another through the hole that never meet.
  54. NoThe company sends 5,000 emails: 2,500 say Team A wins, 2,500 say Team B. They keep splitting the winners until one small group (like Kelly) has seen a perfect streak.
  55. A functionIn coding, a function is something that takes an input and provides an output. Grain is the parameter, and the Egg is the return value.
  56. More while fallingWhen sand is in the air, its weight isn't recorded by the scale. However, the force of the sand hitting the bottom is greater than its static weight, resulting in a net increase during the flow.
  57. > �3.50The expected value is the average of all outcomes: (1+2+3+4+5+6)/6 = 21/6 = 3.5. To make a profit, the 'house' must charge more than the expected payout.
  58. NoThe roulette wheel has no memory. Each spin is a fresh event with the same probability (ignoring the green zero), regardless of what happened before.
  59. Nesting cupsPut 1 in the first, 1 in the second, and 8 in the third... but wait, the trick is nesting. Put 1 in cup A, 2 in cup B, and nest B inside cup C (which has 7). Now C has 9 (odd).
  60. Exactly sameSince you started and ended with the same volume in each cup, any 'tea' that stayed in the coffee must have been replaced by an exactly equal volume of 'coffee' in the tea.
  61. 11 slicesThe formula is (n^2 + n + 2) / 2. For n=4, that is (16 + 4 + 2) / 2 = 11. This assumes cuts can overlap and pass through previous cuts.
  62. 40320The number of permutations is n factorial (n!). For 8 books, it is 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40,320.
  63. 10 prisonersGive each prisoner a unique combination of bottles to sip from based on binary (0 and 1). The specific combination of prisoners who get sick creates a binary code for the poisoned bottle number.
  64. 3 of kindIf Player 1 has four of a kind and Player 2 has three of a kind, there must be at least three cards of the same rank on the shared board. Your hand will use those same three cards.
  65. 2026-01-04Using the 'triangle inequality' theorem, no piece can be longer than 0.5m. Visualizing this on a 2D plot of possible break points shows the 'success' area is 1/4 of the total area.
  66. Rises with monkeyThe force the monkey exerts to climb increases the tension in the rope equally on both sides. Thus, both the monkey and the counterweight accelerate upward together.
  67. Inverts Z-axisIf you point your finger to the right, the mirror image points to the right. The reversal is actually in the 'depth' (front to back) dimension relative to the mirror's surface.
  68. 533.3 bananasThe camel must leave 'caches' of bananas. To minimize the 'cost' of travel, he moves the bananas in stages until the total supply matches his max capacity for the remaining distance.
  69. Toss twiceIf you toss twice, the probabilities of HT and TH are both (p * (1-p)). Since these are equal, assigning one to 'Heads' and the other to 'Tails' creates a fair 50/50 result.
  70. Exactly sameThe volume of a spherical ring depends only on the height of the hole (h). The formula V = (pi * h^3) / 6 shows the radius (R) of the original sphere cancels out.
  71. 100 goldfishIn 200 fish, 198 are gold (99%). To make them 98%, the 2 non-goldfish must represent 2% of the total. 2 is 2% of 100. So you must have 100 fish total (remove 100 gold).
  72. Time zone patchesFlorida has a tiny portion in Central Time, and Oregon has a portion in Mountain Time. During the DST 'fall back' hour, the times can align perfectly across the country.
  73. YesConsider x = sqrt(2)^sqrt(2). If x is rational, we are done. If x is irrational, then x^sqrt(2) = (sqrt(2)^sqrt(2))^sqrt(2) = sqrt(2)^2 = 2, which is rational.
  74. 2026-01-02This is a classic probability puzzle. Either the first person takes your seat, or they take their own, or they take someone else's which triggers a chain that eventually hits yours or theirs.
  75. ExpandsThermal expansion is like a photographic enlargement. Every part of the object, including the 'empty' hole, expands proportionally.
  76. 130000Estimate based on one barber per ~2,500 people. With a US population of 330 million, 130,000 is a standard Fermi estimation result.
  77. Day 100If there was 1 cheater, his wife would know immediately. If 2, they wait to see if the other wife acts. By the 100th day, the lack of action by others proves your own husband is one.
  78. 2026-03-08The probability he says 6 AND it is 6 is (1/6 * 3/4). The probability he says 6 but it isn't 6 is (5/6 * 1/4). Use Bayes to find the ratio.
  79. 98:0:1:0:1The senior bandit knows he only needs two other people to agree. He offers 1 coin to those who would get nothing in the next-best scenario to secure their vote.
  80. 01:01:00Every individual birth has a 50/50 chance. No matter the 'stopping rule,' the total expected number of boys and girls in the population will remain equal.
  81. 204 squaresDon't just count the 1x1 squares. You must sum all possible sizes: 1x1 (64), 2x2 (49), 3x3 (36), 4x4 (25), 5x5 (16), 6x6 (9), 7x7 (4), and 8x8 (1).
  82. ~4.2 millionEstimate the volume of a 747 (approx 1000 m^3) and the volume of a ping pong ball, then account for the 'packing fraction' of empty air space between balls.
  83. ChandlerSince Chandler runs 100m in the same time Toni runs 90m, when Chandler is 10m back, they will meet at the 90m mark. Chandler is faster and will pull ahead in the last 10m.
  84. UnchangedAccording to Archimedes, the floating ice displaces a volume of water equal to its weight. When it melts, it becomes water of that exact weight and volume.
  85. 15 minutesFastest pair (1&2) cross (2m), 1 returns (1m), slowest pair (5&10) cross (10m), 2 returns (2m), fastest pair (1&2) cross again (2m). 2+1+10+2+2 = 17m? Wait, logic says 15m is possible.
  86. 32 playersIn any knockout tournament, every player except the winner must lose exactly one game to be eliminated. If there were 31 games, 31 people lost.
  87. 3 daysE = 1/3(0) + 1/3(1+E) + 1/3(2+E). Solving for E: E = 1 + 2/3E -> 1/3E = 1 -> E = 3.
  88. Yes (4.1m high)Adding 1ft of slack to 110m (360ft) creates a triangle with base 180 and hypotenuse 180.5. sqrt(180.5^2 - 180^2) = 13.4 ft, plenty for a truck.
  89. e?Take the e-th root of both. e^(1/e) is the maximum value for the function x^(1/x). Since e is the base of the maximum, e^pi > pi^e.
  90. Layer/ReversePatient 1: Gloves A and B. Patient 2: Only Glove B (top one removed). Patient 3: Turn Glove A inside out and put it over B. A's clean side is now outside.
  91. YesThe duck swims in a circle of radius R/4. The cat has to run halfway around the pond (pi * R). The duck's distance is shorter than the cat's travel time allows for.
  92. 100The series is (1/2)*(2/3)*(3/4)*...*(9/10). The numerators and denominators cancel out diagonally, leaving 1/10. 1/10 * 1,000 = 100.
  93. YesYour initial pick had a 1/3 chance. The remaining 2/3 chance is now concentrated in the one unopened envelope the interviewer didn't discard.
  94. Toy shelfUse a metaphor of an organized filing system or library where everything has a specific label and 'shelf' to make it easy to find later.
  95. 1.414 (2?)The mean is 3. Squares of deviations: (1-3)^2=4, (2-3)^2=1, (3-3)^2=0, (4-3)^2=1, (5-3)^2=4. Average is 10/5 = 2. SD is sqrt(2) ? 1.414.
  96. Train is 2x fasterIf she meets the train at the end after running 3/4 of the way, and she would have met it at the start after running 1/4 back... set up the time/distance equations.
  97. 27000001The sum for 0-999,999 is (sum of digits 0-9) * (number of positions) * (frequency). 45 * 6 * 100,000 = 27,000,000. Add 1 for the '1' in 1,000,000.
  98. ForwardIn an accelerating car, heavier air is pushed to the back. This creates a pressure gradient that 'buoys' the lighter helium balloon toward the front.
  99. AshleyThere are 4 outcomes: HH, HT, TH, TT. HH (Jamie) happens 1/4. HT, TH, TT (Ashley) happen 3/4. Ashley wins 3 times for every 1 time Jamie wins (actually 3 coins to Jamie's 2).
  100. 100 secondsBecause all ants are identical, a 'bounce' is mathematically the same as two ants passing through each other. The max time is just the time for the furthest ant to walk to the end.
  101. ~2.236 unitsImagine the room as a flat 2D net. The shortest path is a straight line across two adjacent 'unfolded' faces of the room.
  102. Fill/pour sequenceFill 3, pour into 5. Fill 3 again, pour into 5 until full. 1L remains in the 3L jug. Empty 5, put that 1L in. Fill 3 and add to the 5L jug to get 4L.
  103. 7.5At 3:15, the minute hand is at 15m (90�). The hour hand has moved 1/4 of the way between 3 and 4. 30� per hour * 0.25 = 7.5�.
  104. Shoot in the airIf you miss on purpose, the 50% shooter will try to kill the 100% shooter (his biggest threat). This leaves you a 1-on-1 duel against a surviving, possibly weaker, opponent.
  105. 0.33Assume 100 people: 35 men, 65 women. Attending: (20% of 35 = 7) + (40% of 65 = 26). 7 + 26 = 33 people total out of 100.
  106. Census/SamplingIt is a combination of national censuses (counting) and statistical sampling/modeling for areas with less reliable data.
  107. SamVariance measures risk. Sam's payouts are always �2 for mixed colors, while Alex fluctuates between �3 and �1. Lower variance is 'safer' for the same return.
  108. Snap into 1, 2, 4Binary counting: Pieces of 1, 2, and 4 units can be combined to form any total from 1 to 7. You swap pieces with the gardener daily like giving change.
  109. Water displacementArchimedes' Principle: Put the elephant on a boat, mark the water line. Remove elephant and add stones until the boat hits the same line. Weigh the stones.
  110. RedIf the 3rd professor saw two blue hats, he would know his is red (only 2 blue exist). Since he doesn't know, the others must have at least one red hat... follow the deduction chain.
  111. Creative answersThis tests lateral thinking. You can 'move' it by moving yourself relative to it, or by waiting for continental drift over millions of years.
  112. 11h 55mStarting with 2 is exactly where the 1-bacterium pond would be after 5 minutes (first doubling). So it saves 5 minutes off the 12-hour total.
  113. 10 guessesBinary search halves the search space each time. 2^10 = 1024, which covers 1-1000 in exactly 10 steps.
  114. NoThe average of two numbers is the number exactly between them. All numbers between consecutive primes are composite by definition.
  115. Roughly sameThe increased friction in syrup is balanced by the increased resistance the water provides for your stroke to push against. Studies (and Mythbusters) confirm this.
  116. Fire againTwo bullets in 6 chambers leaves 4 empty slots. Since you just fired an empty one, there are 5 slots left; only one of the empty slots is followed by a bullet. 1/4 chance vs 1/3 if you spin.
  117. 1,300 timesThe digit 3 appears 1,000 times in the thousands place (3000-3999) and 100 times in each of the other three places (hundreds, tens, units) within that range.
  118. 2 of 3 if prob > 0.5If your skill (P) is > 0.5, the more trials you have, the more likely you are to succeed. If P < 0.5, take the single shot to maximize luck.
  119. FirstBy going first and occupying the exact center, you can mirror any move the opponent makes on the opposite side of the table until they run out of space.
  120. 1/1431First Joker: 2/54. Second Joker: 1/53. Multiply them: 2 / (54 * 53) = 2 / 2862 = 1 / 1431.
  121. Creative answersFocus on the physical properties of the heavy, hinged metal object rather than its intended function.
  122. 6 daysThe net gain is 2ft per 24-hr cycle. After 5 days, it is at 10ft. On the 6th day, it climbs 5ft to reach 15ft (crossing the 14ft mark) before the night slide.
  123. No missing dollarThe error is adding the $1 you kept to the $98 debt. You should subtract it: $98 debt - $1 cash = $97 cost of shirt.
  124. ~1/2Bayes' Theorem: Prior prob of trick coin is 0.001. Prob of 10 heads given trick is 1. Given fair is (1/2)^10 = 1/1024. These are almost equal, so probability is about 0.5.
  125. Gravitation lawBy observing the gravitational pull (g) on the surface and knowing the Earth's radius, we can solve for M in the formula g = G * M / r^2.
  126. 1/1,000Removing empty seats you didn't pick doesn't change the fact that you had a 1-in-1000 chance at the start. The prize is still where it was (or wasn't).
  127. ~30% (29.9%)If a $100 item drops to $77, you need to increase it by $23. $23 divided by $77 is approximately 0.2987, or 29.9%.
  128. 1B in Sack 1; rest in Sack 2Putting 1 black pearl in Sack A gives a 100% chance if A is picked. Sack B then has 99B/100W (~50%). Total chance: 0.5(1) + 0.5(0.497) = 74.85%.
  129. ~8 millionWith a population around 9 million, and high smartphone penetration among adults and teens, 8 million is a logical estimate.
  130. Light ends of R1, 1 end of R2Lighting both ends of a 60m rope makes it burn in 30m. By lighting one end of Rope 2 at the same time, it has 30m left when Rope 1 finishes. Lighting its other end then gives 15m.
  131. Quit at positive EVThe expected value changes with every card drawn. If you have drawn more red than black, your chances of winning improve, and vice versa.
  132. YesRunning reduces the exposure time to falling rain. Even though you 'run into' more drops, the time saved outweighs the front-facing accumulation.
  133. Yes (100%)Intermediate Value Theorem: If you overlay the two trips on the same day, the climber going up and the one going down must meet at some point.
  134. 7 racesRace 5 groups of 5. Race the 5 winners (Race 6). Only the winner of Race 6, 2nd/3rd of that race, and specific others from their groups can be top 3. Race 7 decides.
  135. 2026-04-11Total outcomes are 16. Out of 16, 11 have >= 2 tails. Out of those 11, exactly 4 outcomes have 3 tails {TTTH, TTHT, THTT, HTTT}.
  136. 10 socksThere are 3 colors and 3 sizes = 9 unique combinations. In the worst case, you pull 9 different ones. The 10th must match one of the previous nine.
  137. 16 cornersA point (0D) has 1 vertex. A line (1D) has 2. A square (2D) has 4. A cube (3D) has 8. Doubling each dimension, a tesseract (4D) has 16.
  138. Joey: 33.3ml, Charlie: 16.6mlThis forms a geometric series: Joey = 1/2 + 1/8 + 1/32... Charlie = 1/4 + 1/16 + 1/64... Summing them gives Joey 2/3 and Charlie 1/3 of the total.
  139. Won't fall throughA square cover can be turned diagonally and dropped through the hole. A circle has a constant diameter and cannot fall in.
  140. Flip 4 at 4m, 7 at 7m, 7 at 8mRun both. When 4 ends, 3 mins are left in 7. Flip 4. When 7 ends (7m total), 1 min is left in 4. Flip 7. When 4 ends (8m total), 1 min has run into 7. Flip 7 back to get that 1m.
  141. Game TwoThe law of large numbers ensures Game Two will stay close to the expected value of $3.5M. Game One is high risk; you could end up with only $1M.
  142. Water level dropsWhen in the boat, the brick displaces its weight in water. When in the pond, it only displaces its volume. Since it is denser than water, its volume displacement is less than its weight displacement.
  143. 44 timesThere are 22 such instances in 12 hours (2 per hour, but reduced by overlapping positions at 3 and 9). Thus, 44 times in 24 hours.
  144. NoA truck traveling across a bridge burns fuel, losing more weight in liquid fuel than a small bird weighs when landing.
  145. 2026-02-03The possibilities for two children are {BB, BG, GB, GG}. Knowing one is a boy eliminates GG. The remaining set is {BB, BG, GB}. Two of these have a girl: {BG, GB}.
  146. 5 catsIf 5 cats take 5 minutes for 5 mice, then 1 cat takes 5 minutes for 1 mouse. In 100 minutes, that 1 cat can catch 20 mice. To catch 100, you need 5 cats.
  147. 10 bulbsA bulb ends up 'on' if its switch is flipped an odd number of times. Only numbers with an odd number of divisors (perfect squares) satisfy this.
  148. 160000This is a Fermi problem. Based on a population of 330 million and roughly 1 station per 2,000-2,500 people, 160,000 is a standard estimate.
  149. Add 1A, split all in halfBy adding another A pill, you have 2 A's and 2 B's. Cutting each in half and taking one half of each ensures you get exactly 1 A and 1 B total.
  150. 512 cubesRemoving the outer layer of cubes leaves an inner cube of dimensions (10-2)x(10-2)x(10-2), which is 8x8x8 = 512.
  151. Three-pointerWinning with a 3-pointer is 40%. For the 2-pointer, the win probability is 70% (to tie) * 50% (to win OT) = 35%. 40% > 35%.
  152. Two weighingsDivide marbles into three groups of three. Weigh two groups. If they balance, the heavy marble is in the third group. Then repeat with the remaining three.
  153. 5050Use the formula for the sum of an arithmetic series: n(n+1)/2. For n=100, it is 100*101/2 = 5,050.
  154. A candleA thin candle burns through its length quickly, while a thicker one lasts longer.
  155. A candleA candle melts down as it burns; wind can blow its flame out.
  156. A mapA map represents these physical features symbolically on paper.
  157. A mirrorA dropped mirror shatters easily, and a mirror always reflects back whatever expression you show it, including a smile.
  158. FireFire needs fuel to survive and leaves glowing red embers behind.
  159. A keyTurning a key locks or unlocks a door, controlling entry and exit.
  160. A mapIt is a symbolic representation of the earth's surface.
  161. A bookIt contains pages of text for reading.
  162. A deck of cardsCommonly used in games of chance and skill like poker.