68 brain teasers for adults with answers

Expect a plausible trap or several linked steps.

1. Starting at 4, alternate multiplying by 2 and subtracting 1. Which term follows 4, 8, 7, 14, 13?

   26. The operations alternate between multiplying by 2 and subtracting 1.The solver must test and retain two alternating operations.Source: OpenStax, College Algebra 2e, Sequences

2. A sequence alternates two streams: one counts 2, 4, 6 and the other 10, 20, 30. What comes after 2, 10, 4, 20, 6, 30?

   8. The odd-position terms are 2, 4, 6, 8 while the even-position terms are 10, 20, 30.Two interleaved progressions must be separated and tracked.Source: OpenStax, College Algebra 2e, Sequences

3. The differences in 2, 5, 10, 17, 26 are consecutive odd numbers. Complete the sequence.

   37. Add consecutive odd numbers 3, 5, 7, 9, then 11.The first differences must be calculated before their pattern appears.Source: OpenStax, College Algebra 2e, Sequences

4. The amount added grows by 3 in 1, 2, 6, 13, 23. What is the next term?

   36. The additions are 1, 4, 7, 10, then 13.The solver needs a difference list and a second pattern within it.Source: OpenStax, College Algebra 2e, Sequences

5. Odd positions count upward by 1 and even positions fall by 10 in 1, 100, 2, 90, 3, 80. Which term is seventh?

   4. The odd positions count upward while the even positions fall by 10.The prompt signals two tracks, but both must still be checked.Source: OpenStax, College Algebra 2e, Sequences

6. Two streams are woven together: one adds 3 and the other subtracts 2. Give the next term after 5, 20, 8, 18, 11, 16.

   14. One stream adds 3 while the other subtracts 2.The two position-based rules require modest bookkeeping.Source: OpenStax, College Algebra 2e, Sequences

7. Forward gaps grow by one letter at a time in B, E, I, N. Which letter is next?

   T. The forward gaps are 3, 4, 5, then 6 letters.Letter positions and changing differences must both be tracked.Source: OpenStax, College Algebra 2e, Sequences

8. Backward gaps grow by one letter at a time in Z, W, S, N. Where do you land next?

   H. Move back 3, then 4, then 5, then 6 letters.The solver must translate letters to ordered positions and extend the gap pattern.Source: OpenStax, College Algebra 2e, Sequences

9. Triangular numbers begin 1, 3, 6, 10, 15. Continue the pattern once.

   21.The changing additions must be identified before the next term can be calculated.Source: OpenStax, College Algebra 2e, Sequences

10. Successive products form 1, 2, 6, 24, 120 by multiplying by 2, 3, 4, then 5. Extend the pattern.

    720.The multiplier changes at every step and must be advanced correctly.Source: OpenStax, College Algebra 2e, Sequences

11. Terms follow n times the next whole number: 2, 6, 12, 20, 30 for n from 1 through 5. Find the term for n = 6.

    42.The stated formula must be matched to the examples and evaluated once.Source: OpenStax, College Algebra 2e, Sequences

12. Beginning at 12, add the digits of the current term to make the next: 12, 15, 21, 24, 30. Continue once.

    33.The amount added depends on each changing term rather than on a fixed difference.Source: OpenStax, College Algebra 2e, Sequences

13. Pair each counting number with twice itself: 1, 2, 2, 4, 3, 6, 4, 8. Which term is ninth?

    5.The sequence must be regrouped into ordered pairs before it can be extended.Source: OpenStax, College Algebra 2e, Sequences

14. Four printers make four posters in four minutes. At the same individual rate, how long do 20 printers need to make 20 posters?

    Four minutes.A plausible multiplication trap must be rejected by finding the per-printer rate.Source: OpenStax, Prealgebra 2e, Proportions

15. Six taps each fill one identical bucket at the same time. If six buckets fill in 12 minutes, how long would one tap take to fill one bucket?

    12 minutes.The solver must distinguish parallel work from sequential work.Source: OpenStax, Prealgebra 2e, Proportions

16. Take the first of four tablets now and another every 20 minutes. When is the last tablet taken?

    60 minutes from now.The first event occurs at time zero, which creates a common off-by-one trap.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

17. Ava says, "Bo is lying." Bo says, "Ava and I are the same type." One always tells the truth and one always lies. Who is who?

    Ava tells the truth and Bo lies.Both possible assignments must be tested for consistency.Source: Open Logic Project, forall x: Calgary

18. Two fathers and two sons share three oranges so that each person receives one. How many people are there?

    Three: a grandfather, his son, and his grandson.Overlapping family roles must replace the default assumption of four people.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

19. Facing north, turn right, right, left, then around. Which direction are you facing?

    West.Four orientation updates must be tracked without losing the current direction.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

20. Three closed boxes are labeled APPLES, ORANGES, and MIXED, but every label is wrong. From which box should one fruit be drawn to relabel all three?

    Draw from the box labeled MIXED.The all-wrong condition must be used to turn one observation into three forced labels.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

21. One word follows rain, sun, and cross to make three familiar compounds. What is it?

    BOW: rainbow, sunbow, and crossbow.Three separate compounds must accept the same word.Source: Merriam-Webster.com Dictionary, "rainbow"

22. Base, basket, and foot can each take the same ending. Supply it.

    BALL: baseball, basketball, and football.Three separate starting words must be tested against one shared ending.Source: Merriam-Webster.com Dictionary, "baseball"

23. Change one letter at a time from COLD to WARM, making a word after every change. Give a four-move route.

    COLD, CORD, CARD, WARD, WARM.Four linked substitutions must keep every intermediate state valid.Source: Merriam-Webster.com Dictionary, "cold"

24. Find a body organ hidden across the words "the artist."

    HEART.The answer crosses a word boundary and is not signaled by spacing.Source: Merriam-Webster.com Dictionary, "heart"

25. Seven people each shake hands once with every other person. How many handshakes occur?

    21 handshakes.Pairs must be counted without double-counting.Source: OpenStax, College Algebra, Counting Principles

26. After a 25 percent discount, a game costs $45. What was its original price?

    $60.The solver must treat $45 as 75 percent rather than subtracting 25 percent again.Source: OpenStax, Prealgebra 2e, Proportions

27. A price rises by 20 percent and becomes $72. Find the price before the increase.

    $60.The final amount must be recognized as 120 percent of the original.Source: OpenStax, Prealgebra 2e, Proportions

28. Two whole numbers total 42 and differ by 10. What are they?

    26 and 16.Two conditions must be represented and solved together.Source: OpenStax, Prealgebra 2e, Solving Equations

29. A square has area 121 square units. What is its perimeter?

    44 units.The side length must first be recovered before the perimeter is found.Source: OpenStax, Prealgebra 2e, Rectangles, Triangles, and Trapezoids

30. Two independent fair coins are tossed. What is the probability of getting exactly one head?

    1/2.All equally likely paired outcomes must be listed without omission.Source: OpenStax, Prealgebra 2e, Averages and Probability

31. Name the smallest positive whole number divisible by 4, 6, and 9.

    36.Three factor requirements must be combined into one least common multiple.Source: OpenStax, College Algebra, Counting Principles

32. Which is the largest three-digit multiple of 7?

    994.A boundary value must be divided, rounded down, and checked.Source: OpenStax, College Algebra, Counting Principles

33. Find the sum of the whole numbers from 1 through 20.

    210.The solver must recognize or construct paired terms rather than add blindly.Source: OpenStax, College Algebra 2e, Sequences

34. A positive whole number below 20 leaves remainder 2 when divided by 3 and remainder 4 when divided by 5. Find it.

    14.Two remainder conditions must be intersected over a bounded set.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

35. Five quiz scores have mean 18. If the score 14 is removed, what is the mean of the other four?

    19.The original total must be reconstructed before a new mean can be calculated.Source: OpenStax, Prealgebra 2e, Averages and Probability

36. Traveling at a constant 18 kilometers per hour, how far does a cyclist go in 50 minutes?

    15 kilometers.Minutes must be converted to a fraction of an hour before applying the rate.Source: OpenStax, Prealgebra 2e, Proportions

37. A bag contains 3 red and 2 blue counters. Two are drawn without replacement. What is the probability both are red?

    3/10.The second probability must use the reduced counts after the first draw.Source: OpenStax, Prealgebra 2e, Averages and Probability

38. In a group, 18 people drink tea, 13 drink coffee, and 7 drink both. How many drink at least one of the two?

    24 people.The overlap must be removed once after it is counted in both totals.Source: OpenStax, College Algebra, Counting Principles

39. Five runners finish in distinct places. Quin is first, Noor is last, Lia is immediately before Mo, and Pax is before Lia. Give the order.

    Quin, Pax, Lia, Mo, Noor.A linked two-person block and two fixed endpoints force several placements.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

40. Ana, Bo, and Cy each have a different pet: cat, dog, or fish. Ana does not have the cat, Bo has the dog, and Cy does not have the fish. Which pet does Ana have?

    The fish.Two eliminations across three one-to-one assignments are required.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

41. Three cups contain tea, juice, and water, one drink per cup. Blue has water, red does not have tea, and green does not have juice. What is in red?

    Juice.The fixed assignment must be removed before two negative clues resolve the rest.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

42. In a reversed-alphabet code, A pairs with Z, B with Y, and so on. Decode WLT.

    DOG.Three non-adjacent mirror positions must be translated accurately.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

43. All metal widgets are heavy. Widget X is not heavy. What follows?

    Widget X is not a metal widget.The solver must use the contrapositive of a universal rule.Source: Open Logic Project, forall x: Calgary

44. All tulips are flowers, and some flowers are yellow. Must some tulips be yellow?

    No.A tempting overlap is not guaranteed by the two set statements.Source: Open Logic Project, forall x: Calgary

45. Every club member who swims goes without a hat. Ivo is a club member and wears a hat. Can Ivo be a swimmer under the rule?

    No.The stated observation must be combined with a universal conditional.Source: Open Logic Project, forall x: Calgary

46. Find the number greater than 40 and less than 50 that is even, divisible by 3, and has digits summing to 6.

    42.Several filters must be intersected over a small candidate set.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

47. A two-digit number has a tens digit one greater than its ones digit, and its digits total 9. What is the number?

    54.Two digit constraints must be solved together.Source: OpenStax, Prealgebra 2e, Solving Equations

48. Tasks W, X, Y, and Z must be done once each. W is before X, Y is after X, and Z is before W. Give the only order.

    Z, W, X, Y.Three precedence clues must be joined into one four-step chain.Source: Open Logic Project, forall x: Calgary

49. Talks J, K, L, and M fill slots 1 through 4. M is third, K is after M, L is before J, and L is not second. Give the order.

    L, J, M, K.Fixed placement, relative order, and exclusion clues must be combined.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

50. Exactly two of lamps A, B, C, and D are on. A and B have opposite states, C and D have opposite states, A matches C, and D is on. Which lamps are on?

    B and D.Four linked binary constraints must be propagated without switching their direction.Source: Open Logic Project, forall x: Calgary

Allow working time and ask the solver to explain the route.

1. Describe each term aloud to obtain the next: 1, 11, 21, 1211, 111221. What follows?

   312211.The non-arithmetic rule must be inferred and applied to three separate runs of digits.Source: OpenStax, College Algebra 2e, Sequences

2. Nine coins begin tails-up. One move flips exactly two coins. Can repeated moves leave all nine heads-up?

   No.The decisive parity invariant is not obvious from trying individual moves.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

3. Remove two opposite corner squares from an 8 by 8 checkerboard. Can 31 dominoes, each covering two edge-sharing squares, cover what remains?

   No. Opposite corners have the same color, while every domino covers one square of each color.A non-obvious color invariant proves impossibility without exhaustive placement.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

4. Box 1 says, "The prize is here." Box 2 says, "The prize is not here." Exactly one label is said to be true. Can either box hold the prize without breaking that rule?

   No.The solver must exhaust both placements and notice that each makes the labels agree.Source: Open Logic Project, forall x: Calgary

5. Nine identical-looking coins include one heavier coin. How can a balance scale identify it in two weighings?

   Weigh three against three. Keep the heavier group, or the unweighed group if they balance. Then weigh two coins from that group against each other.A three-way outcome must be preserved at each of two planned tests.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

6. Four people take 1, 2, 7, and 10 minutes to cross a bridge. At most two cross, they need one torch, and a pair moves at the slower person's speed. What is the least total time?

   17 minutes.Several linked crossings must be optimized and lower-cost alternatives compared.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

7. Three switches control one indicator in another room. The indicator stays warm for five minutes after it is switched off. You may enter the room once. How can you identify the controlling switch?

   Turn on switch 1 for several minutes, turn it off, turn on switch 2, then enter. Lit means switch 2, warm but unlit means switch 1, and cool and unlit means switch 3.The solution must encode three states using both current and stored information.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

8. Two unevenly burning ropes each take exactly 60 minutes to burn from one end to the other. How can they measure 45 minutes?

   Light the first rope at both ends and the second rope at one end. When the first is gone, light the other end of the second.The method must avoid assuming that either rope burns at a uniform rate.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

9. Using only an unmarked 3-liter jug and an unmarked 5-liter jug, measure exactly 4 liters.

   Leave 2 liters in the 5-liter jug, transfer them to the empty 3-liter jug, then fill the 5-liter jug and pour 1 liter into the smaller jug.Several fills, pours, and empties must preserve an intermediate quantity.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

10. Eight numbered vials include exactly one contaminant that darkens a test strip. How can three strips identify the vial in one test round?

    Number the vials 0 through 7 in three-bit binary and put a drop on each strip corresponding to a 1 in the vial's code.Eight possibilities must be encoded into the eight outcome patterns of three binary tests.Source: OpenStax, College Algebra, Counting Principles

11. A traveler must ferry a fox, a hen, and grain across a river, carrying only one of them at a time. The fox cannot stay with the hen, and the hen cannot stay with the grain. Give a safe crossing plan.

    Take the hen across, return alone, take the fox across, bring the hen back, take the grain across, return alone, then take the hen across.The plan requires two deliberate returns while maintaining both safety constraints after every trip.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

12. One hundred closed lockers are toggled on pass 1, every second locker on pass 2, every third on pass 3, and so on through pass 100. Which lockers remain open?

    The ten lockers numbered by perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.A hidden divisor-count invariant replaces an impractical simulation of all passes.Source: OpenStax, College Algebra, Counting Principles

13. Build a five-move word ladder from HEAD to TAIL, changing one letter per move.

    HEAD, HEAL, TEAL, TELL, TALL, TAIL.Five linked substitutions require planning through valid intermediate words.Source: Merriam-Webster.com Dictionary, "head"

14. Pages are numbered from 1 through 120. How many times is the digit 1 printed?

    53 times.Overlapping hundreds, tens, and ones cases must be counted systematically.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

15. A rectangle has positive whole-number side lengths and area 36. Which side lengths give the smallest perimeter?

    6 by 6, with perimeter 24.All factor pairs must be compared or bounded to prove the minimum.Source: OpenStax, Prealgebra 2e, Rectangles, Triangles, and Trapezoids

16. A rectangle has positive whole-number side lengths and perimeter 22. Which dimensions give the greatest area?

    5 by 6, with area 30.Every whole-number side pair meeting the perimeter constraint must be compared.Source: OpenStax, Prealgebra 2e, Rectangles, Triangles, and Trapezoids

17. A three-digit code obeys these exact clues: 682 has one digit correct and well placed; 614 has one correct but misplaced; 206 has two correct but misplaced; 738 has none correct; 780 has one correct but misplaced. Find the code.

    042.Five exact-count clues must be intersected while tracking both membership and position.Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills

18. Exactly one of three suspects took a key, and exactly one statement is true. Ana says, "I did not take it." Bo says, "Ana took it." Cy says, "Bo did not take it." Who took the key?

    Bo.All three culprit cases must be checked against an exact truth count.Source: Open Logic Project, forall x: Calgary