68 brain teasers for adults with answers
Medium and hard reasoning challenges for adult groups, with a checked answer and a short account of the work each puzzle requires.
68 puzzles · 68 brain teasers for adults with answers · answers included
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Medium challenges
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?
Hint: The last step was subtracting 1, so the next step is the other operation.
Answer26. The operations alternate between multiplying by 2 and subtracting 1.
4 x 2 = 8, 8 - 1 = 7, 7 x 2 = 14, 14 - 1 = 13, then 13 x 2 = 26.
Source: OpenStax, College Algebra 2e, Sequences https://openstax.org/books/college-algebra-2e/pages/9-1-sequences-and-their-notations
- 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?
Hint: Read every other term. The next term continues the stream that began with 2.
Answer8. The odd-position terms are 2, 4, 6, 8 while the even-position terms are 10, 20, 30.
Positions 1, 3, and 5 rise by 2, so position 7 is 8.
Source: OpenStax, College Algebra 2e, Sequences https://openstax.org/books/college-algebra-2e/pages/9-1-sequences-and-their-notations
- 3
The differences in 2, 5, 10, 17, 26 are consecutive odd numbers. Complete the sequence.
Hint: The gaps so far are 3, 5, 7 and 9. What is the next odd number?
Answer37. Add consecutive odd numbers 3, 5, 7, 9, then 11.
The differences are 3, 5, 7, and 9; the next difference is 11, and 26 + 11 = 37.
Source: OpenStax, College Algebra 2e, Sequences https://openstax.org/books/college-algebra-2e/pages/9-1-sequences-and-their-notations
- 4
The amount added grows by 3 in 1, 2, 6, 13, 23. What is the next term?
Hint: The amounts added so far are 1, 4, 7 and 10.
Answer36. The additions are 1, 4, 7, 10, then 13.
The differences increase by 3: 1, 4, 7, 10, 13. Therefore 23 + 13 = 36.
Source: OpenStax, College Algebra 2e, Sequences https://openstax.org/books/college-algebra-2e/pages/9-1-sequences-and-their-notations
- 5
Odd positions count upward by 1 and even positions fall by 10 in 1, 100, 2, 90, 3, 80. Which term is seventh?
Hint: The seventh term is in an odd position, so follow the stream that counts up.
Answer4. The odd positions count upward while the even positions fall by 10.
Odd positions 1, 3, 5, and 7 contain 1, 2, 3, and 4.
Source: OpenStax, College Algebra 2e, Sequences https://openstax.org/books/college-algebra-2e/pages/9-1-sequences-and-their-notations
- 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.
Hint: The next term belongs to the stream that started at 5.
Answer14. One stream adds 3 while the other subtracts 2.
Odd positions are 5, 8, 11, so the next odd-position term is 14.
Source: OpenStax, College Algebra 2e, Sequences https://openstax.org/books/college-algebra-2e/pages/9-1-sequences-and-their-notations
- 7
Forward gaps grow by one letter at a time in B, E, I, N. Which letter is next?
Hint: The gaps are 3, 4 and 5 letters. Move forward 6 from N.
AnswerT. The forward gaps are 3, 4, 5, then 6 letters.
B to E is 3, E to I is 4, I to N is 5, and six letters after N is T.
Source: OpenStax, College Algebra 2e, Sequences https://openstax.org/books/college-algebra-2e/pages/9-1-sequences-and-their-notations
- 8
Backward gaps grow by one letter at a time in Z, W, S, N. Where do you land next?
Hint: Count back from N by one more letter than last time.
AnswerH. Move back 3, then 4, then 5, then 6 letters.
N is the 14th letter; moving back 6 positions reaches the 8th letter, H.
Source: OpenStax, College Algebra 2e, Sequences https://openstax.org/books/college-algebra-2e/pages/9-1-sequences-and-their-notations
- 9
Triangular numbers begin 1, 3, 6, 10, 15. Continue the pattern once.
Hint: Each step adds one more than the step before: 2, 3, 4, 5.
Answer21.
The additions are 2, 3, 4, and 5. Adding the next amount, 6, to 15 gives 21.
Source: OpenStax, College Algebra 2e, Sequences https://openstax.org/books/college-algebra-2e/pages/9-1-sequences-and-their-notations
- 10
Successive products form 1, 2, 6, 24, 120 by multiplying by 2, 3, 4, then 5. Extend the pattern.
Hint: The next multiplier is 6.
Answer720.
The next multiplier is 6, so the next term is 120 x 6 = 720.
Source: OpenStax, College Algebra 2e, Sequences https://openstax.org/books/college-algebra-2e/pages/9-1-sequences-and-their-notations
- 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.
Hint: Check the rule on the last term you have: 5 times 6 is 30.
Answer42.
For n = 6, the rule gives 6 x (6 + 1) = 6 x 7 = 42.
Source: OpenStax, College Algebra 2e, Sequences https://openstax.org/books/college-algebra-2e/pages/9-1-sequences-and-their-notations
- 12
Beginning at 12, add the digits of the current term to make the next: 12, 15, 21, 24, 30. Continue once.
Hint: The digits of 30 add up to 3.
Answer33.
The shown steps add 1 + 2, then 1 + 5, then 2 + 1, then 2 + 4. The digits of 30 total 3, so 30 + 3 = 33.
Source: OpenStax, College Algebra 2e, Sequences https://openstax.org/books/college-algebra-2e/pages/9-1-sequences-and-their-notations
- 13
Pair each counting number with twice itself: 1, 2, 2, 4, 3, 6, 4, 8. Which term is ninth?
Hint: The terms come in pairs, and the ninth term starts a new pair.
Answer5.
The pairs are (1, 2), (2, 4), (3, 6), and (4, 8). The next pair begins with 5, so the ninth term is 5.
Source: OpenStax, College Algebra 2e, Sequences https://openstax.org/books/college-algebra-2e/pages/9-1-sequences-and-their-notations
- 14
Four printers make four posters in four minutes. At the same individual rate, how long do 20 printers need to make 20 posters?
Hint: How long does one printer take to make one poster?
AnswerFour minutes.
Each printer makes one poster in four minutes, so 20 printers make 20 posters in four minutes.
Source: OpenStax, Prealgebra 2e, Proportions https://openstax.org/books/prealgebra-2e/pages/6-5-solve-proportions-and-their-applications
- 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?
Hint: Each tap fills its own bucket, so the taps do not help one another.
Answer12 minutes.
Each tap is paired with one bucket, so the individual fill time is the observed 12 minutes.
Source: OpenStax, Prealgebra 2e, Proportions https://openstax.org/books/prealgebra-2e/pages/6-5-solve-proportions-and-their-applications
- 16
Take the first of four tablets now and another every 20 minutes. When is the last tablet taken?
Hint: The first tablet is taken now, so count only the gaps between tablets.
Answer60 minutes from now.
The four tablet times are 0, 20, 40, and 60 minutes.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-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?
Hint: Suppose Ava tells the truth, then check whether Bo's sentence fits.
AnswerAva tells the truth and Bo lies.
If Ava is truthful, Bo lies and his same-type claim is false. Reversing the roles makes Bo's same-type claim false even though he would be truthful.
Source: Open Logic Project, forall x: Calgary https://forallx.openlogicproject.org/html/Ch11.html
- 18
Two fathers and two sons share three oranges so that each person receives one. How many people are there?
Hint: One person can be a father and a son at the same time.
AnswerThree: a grandfather, his son, and his grandson.
The middle person is both a father and a son, satisfying all four role descriptions among three people.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-develop-key-problem-solving-skills
- 19
Facing north, turn right, right, left, then around. Which direction are you facing?
Hint: Two right turns face you the opposite way.
AnswerWest.
North becomes east, south, east, then west after the half-turn.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-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?
Hint: Every label is wrong, so the box labeled MIXED holds only one kind of fruit.
AnswerDraw from the box labeled MIXED.
The box labeled MIXED cannot be mixed. If the drawn fruit is an apple, that box is APPLES; the box labeled ORANGES cannot be oranges and cannot now be apples, so it is MIXED, leaving ORANGES for the last box. If the draw is an orange, the same reasoning works with the fruit names exchanged.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-develop-key-problem-solving-skills
- 21
One word follows rain, sun, and cross to make three familiar compounds. What is it?
Hint: Think of what an archer carries.
AnswerBOW: rainbow, sunbow, and crossbow.
Appending BOW forms RAINBOW, SUNBOW, and CROSSBOW.
Source: Merriam-Webster.com Dictionary, "rainbow" https://www.merriam-webster.com/dictionary/rainbow
- 22
Base, basket, and foot can each take the same ending. Supply it.
Hint: All three are sports.
AnswerBALL: baseball, basketball, and football.
Appending BALL forms BASEBALL, BASKETBALL, and FOOTBALL.
Source: Merriam-Webster.com Dictionary, "baseball" https://www.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.
Hint: Start by changing the L in COLD.
AnswerCOLD, CORD, CARD, WARD, WARM.
Each adjacent pair differs in exactly one position, and every listed form is a word.
Source: Merriam-Webster.com Dictionary, "cold" https://www.merriam-webster.com/dictionary/cold
- 24
Find a body organ hidden across the words "the artist."
Hint: Ignore the space and read across both words.
AnswerHEART.
tHE ARTist contains the consecutive letters H, E, A, R, T.
Source: Merriam-Webster.com Dictionary, "heart" https://www.merriam-webster.com/dictionary/heart
- 25
Seven people each shake hands once with every other person. How many handshakes occur?
Hint: Each person shakes 6 hands, but every handshake involves two people.
Answer21 handshakes.
There are 7 x 6 ordered choices, and dividing by 2 gives 21 unordered pairs.
Source: OpenStax, College Algebra, Counting Principles https://openstax.org/books/college-algebra/pages/9-5-counting-principles
- 26
After a 25 percent discount, a game costs $45. What was its original price?
Hint: $45 is 75 percent of the original price.
Answer$60.
0.75 times the original price equals 45, so the original price is 45 / 0.75 = 60.
Source: OpenStax, Prealgebra 2e, Proportions https://openstax.org/books/prealgebra-2e/pages/6-5-solve-proportions-and-their-applications
- 27
A price rises by 20 percent and becomes $72. Find the price before the increase.
Hint: $72 is 120 percent of the old price.
Answer$60.
1.2 times the original price equals 72, and 72 / 1.2 = 60.
Source: OpenStax, Prealgebra 2e, Proportions https://openstax.org/books/prealgebra-2e/pages/6-5-solve-proportions-and-their-applications
- 28
Two whole numbers total 42 and differ by 10. What are they?
Hint: Take 10 off 42 and split what is left in two.
Answer26 and 16.
x + y = 42 and x - y = 10. Adding gives 2x = 52, so x = 26 and y = 16.
Source: OpenStax, Prealgebra 2e, Solving Equations https://openstax.org/books/prealgebra-2e/pages/8-1-solve-equations-using-the-subtraction-and-addition-properties-of-equality
- 29
A square has area 121 square units. What is its perimeter?
Hint: Which number times itself makes 121?
Answer44 units.
The positive square root of 121 is 11, and 4 x 11 = 44.
Source: OpenStax, Prealgebra 2e, Rectangles, Triangles, and Trapezoids https://openstax.org/books/prealgebra-2e/pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids
- 30
Two independent fair coins are tossed. What is the probability of getting exactly one head?
Hint: List the four ways two coins can land.
Answer1/2.
The outcomes are HH, HT, TH, and TT. Two of four have exactly one head, so the probability is 2/4 = 1/2.
Source: OpenStax, Prealgebra 2e, Averages and Probability https://openstax.org/books/prealgebra-2e/pages/5-5-averages-and-probability
- 31
Name the smallest positive whole number divisible by 4, 6, and 9.
Hint: Try multiples of 9 until one also divides by 4 and by 6.
Answer36.
The required prime factors are 2 squared and 3 squared, whose product is 36.
Source: OpenStax, College Algebra, Counting Principles https://openstax.org/books/college-algebra/pages/9-5-counting-principles
- 32
Which is the largest three-digit multiple of 7?
Hint: Divide 999 by 7 and look at the remainder.
Answer994.
999 / 7 is 142 remainder 5, so the largest whole multiplier is 142 and 142 x 7 = 994.
Source: OpenStax, College Algebra, Counting Principles https://openstax.org/books/college-algebra/pages/9-5-counting-principles
- 33
Find the sum of the whole numbers from 1 through 20.
Hint: Pair 1 with 20, 2 with 19, and so on.
Answer210.
Pair 1 with 20, 2 with 19, and so on. Ten pairs each total 21, giving 210.
Source: OpenStax, College Algebra 2e, Sequences https://openstax.org/books/college-algebra-2e/pages/9-1-sequences-and-their-notations
- 34
A positive whole number below 20 leaves remainder 2 when divided by 3 and remainder 4 when divided by 5. Find it.
Hint: List the numbers below 20 that leave 4 when divided by 5.
Answer14.
The numbers below 20 that leave remainder 4 upon division by 5 are 4, 9, 14, and 19. Their remainders upon division by 3 are 1, 0, 2, and 1, so only 14 satisfies both conditions.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-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?
Hint: The five scores add up to 90.
Answer19.
A mean of 18 across five scores gives a total of 5 x 18 = 90. Removing 14 leaves 76, and 76 / 4 = 19.
Source: OpenStax, Prealgebra 2e, Averages and Probability https://openstax.org/books/prealgebra-2e/pages/5-5-averages-and-probability
- 36
Traveling at a constant 18 kilometers per hour, how far does a cyclist go in 50 minutes?
Hint: 50 minutes is five sixths of an hour.
Answer15 kilometers.
Fifty minutes is 50/60 = 5/6 of an hour. Distance equals rate times time, so 18 x 5/6 = 15 kilometers.
Source: OpenStax, Prealgebra 2e, Proportions https://openstax.org/books/prealgebra-2e/pages/6-5-solve-proportions-and-their-applications
- 37
A bag contains 3 red and 2 blue counters. Two are drawn without replacement. What is the probability both are red?
Hint: After one red counter is drawn, how many reds are left, out of how many counters?
Answer3/10.
The first red has probability 3/5. After it is removed, the second red has probability 2/4. Multiplying gives 3/5 x 2/4 = 6/20 = 3/10.
Source: OpenStax, Prealgebra 2e, Averages and Probability https://openstax.org/books/prealgebra-2e/pages/5-5-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?
Hint: The 7 who drink both were counted twice.
Answer24 people.
Adding 18 and 13 counts the 7 people in both groups twice. Subtracting that duplicate count gives 18 + 13 - 7 = 24.
Source: OpenStax, College Algebra, Counting Principles https://openstax.org/books/college-algebra/pages/9-5-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.
Hint: Place Quin and Noor first, then keep Lia and Mo side by side.
AnswerQuin, Pax, Lia, Mo, Noor.
Positions 1 and 5 are fixed. Lia-Mo cannot use 2-3 because Pax must be before Lia, so Pax, Lia, and Mo fill 2, 3, and 4.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-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?
Hint: Bo's pet is settled. Which pets are left for Ana and Cy?
AnswerThe fish.
Bo uses dog. Cat and fish remain; Cy cannot take fish, so Cy has cat and Ana has fish.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-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?
Hint: Blue has the water, and green cannot have juice. What does green have?
AnswerJuice.
Blue uses water. Tea and juice remain; green cannot use juice, so green has tea and red has juice.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-develop-key-problem-solving-skills
- 42
In a reversed-alphabet code, A pairs with Z, B with Y, and so on. Decode WLT.
Hint: W is fourth from the end of the alphabet, so it stands for the fourth letter.
AnswerDOG.
W mirrors D, L mirrors O, and T mirrors G.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-develop-key-problem-solving-skills
- 43
All metal widgets are heavy. Widget X is not heavy. What follows?
Hint: If X were a metal widget, what would it have to be?
AnswerWidget X is not a metal widget.
If X were a metal widget, the premise would make it heavy, contradicting the stated fact.
Source: Open Logic Project, forall x: Calgary https://forallx.openlogicproject.org/html/Ch11.html
- 44
All tulips are flowers, and some flowers are yellow. Must some tulips be yellow?
Hint: The yellow flowers might all be daffodils.
AnswerNo.
The yellow flowers could all be non-tulips, so the premises allow a case with no yellow tulips.
Source: Open Logic Project, forall x: Calgary https://forallx.openlogicproject.org/html/Ch11.html
- 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?
Hint: Under the rule, a member who swims never wears a hat.
AnswerNo.
Swimming would imply no hat, which conflicts with Ivo wearing a hat.
Source: Open Logic Project, forall x: Calgary https://forallx.openlogicproject.org/html/Ch11.html
- 46
Find the number greater than 40 and less than 50 that is even, divisible by 3, and has digits summing to 6.
Hint: Only one number in the 40s has digits that add up to 6.
Answer42.
The multiples of 3 in the range are 42, 45, and 48. Only 42 is even and has digit sum 6.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-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?
Hint: Find two neighboring digits that add up to 9.
Answer54.
If the ones digit is x, the tens digit is x + 1. Then 2x + 1 = 9, so x = 4 and the number is 54.
Source: OpenStax, Prealgebra 2e, Solving Equations https://openstax.org/books/prealgebra-2e/pages/8-1-solve-equations-using-the-subtraction-and-addition-properties-of-equality
- 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.
Hint: Z comes before W, and W before X. Where can Y go?
AnswerZ, W, X, Y.
Z precedes W, W precedes X, and X precedes Y, so the full order is forced.
Source: Open Logic Project, forall x: Calgary https://forallx.openlogicproject.org/html/Ch11.html
- 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.
Hint: M is third and K comes after M, so K takes slot 4.
AnswerL, J, M, K.
M occupies slot 3, so K must be slot 4. Slots 1 and 2 remain for L and J; because L is before J and is not second, L is slot 1 and J is slot 2.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-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?
Hint: D is on, so C is off, and A matches C.
AnswerB and D.
D is on, so its opposite C is off. A matches C, so A is off. B is opposite A, so B is on. Therefore exactly B and D are on, satisfying the stated total of two.
Source: Open Logic Project, forall x: Calgary https://forallx.openlogicproject.org/html/Ch11.html
Hard challenges
Allow working time and ask the solver to explain the route.
- 51
Describe each term aloud to obtain the next: 1, 11, 21, 1211, 111221. What follows?
Hint: Read 111221 aloud in groups: three 1s, two 2s, one 1.
Answer312211.
The last shown term has three 1s, two 2s, and one 1. Writing those counts and digits in order gives 31 22 11, or 312211.
Source: OpenStax, College Algebra 2e, Sequences https://openstax.org/books/college-algebra-2e/pages/9-1-sequences-and-their-notations
- 52
Nine coins begin tails-up. One move flips exactly two coins. Can repeated moves leave all nine heads-up?
Hint: Each move changes the number of heads by 0 or by 2. Is 9 even or odd?
AnswerNo.
The number of heads changes by 2, 0, or -2, so it always stays even. Nine heads is odd.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-develop-key-problem-solving-skills
- 53
Remove two opposite corner squares from an 8 by 8 checkerboard. Can 31 dominoes, each covering two edge-sharing squares, cover what remains?
Hint: Think about the colors. Which color are the two corners you removed?
AnswerNo. Opposite corners have the same color, while every domino covers one square of each color.
The board starts with 32 squares of each color. Removing same-colored corners leaves 30 of one color and 32 of the other, but 31 dominoes would cover 31 of each.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-develop-key-problem-solving-skills
- 54
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?
Hint: Put the prize in box 1, then in box 2, and count the true labels each time.
AnswerNo.
With the prize in box 1, both labels are true. With it in box 2, both labels are false.
Source: Open Logic Project, forall x: Calgary https://forallx.openlogicproject.org/html/Ch11.html
- 55
Nine identical-looking coins include one heavier coin. How can a balance scale identify it in two weighings?
Hint: Split the coins into three groups of three.
AnswerWeigh three against three. Keep the heavier group, or the unweighed group if they balance. Then weigh two coins from that group against each other.
The first weighing narrows nine candidates to three. The second distinguishes left, right, or the unweighed third coin.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-develop-key-problem-solving-skills
- 56
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?
Hint: Send the two slowest people across together, and let the two fastest carry the torch back.
Answer17 minutes.
Crossing the two slowest together costs at least 10. The two fastest can enable that crossing and restore the torch for 2 + 1 + 10 + 2 = 15, after which their final crossing costs 2. Sending the slowest separately costs at least 21, so 17 is minimal.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-develop-key-problem-solving-skills
- 57
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?
Hint: The indicator can tell you more than on or off. Touch it.
AnswerTurn 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 procedure deliberately creates three distinguishable states: on, recently off, and never on.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-develop-key-problem-solving-skills
- 58
Two unevenly burning ropes each take exactly 60 minutes to burn from one end to the other. How can they measure 45 minutes?
Hint: A rope lit at both ends burns out in half the time.
AnswerLight 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.
Burning the first rope from both ends consumes it in 30 minutes even if its rate varies along its length. The second rope then has 30 minutes of one-end burn time left; lighting its other end halves that remaining time to 15 minutes. The total is 30 + 15 = 45 minutes.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-develop-key-problem-solving-skills
- 59
Using only an unmarked 3-liter jug and an unmarked 5-liter jug, measure exactly 4 liters.
Hint: Fill the 5-liter jug and pour it into the 3-liter jug. What is left?
AnswerLeave 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.
Fill the 5-liter jug and pour into the 3-liter jug, leaving 2 liters. Empty the smaller jug and move those 2 liters into it. Refill the 5-liter jug; the smaller jug needs 1 more liter, so filling it leaves exactly 4 liters in the larger jug.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-develop-key-problem-solving-skills
- 60
Eight numbered vials include exactly one contaminant that darkens a test strip. How can three strips identify the vial in one test round?
Hint: Three strips that each turn dark or stay clear give 2 times 2 times 2 patterns.
AnswerNumber 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.
Three strips have 2 x 2 x 2 = 8 light-or-dark patterns. Treating dark as 1 and unchanged as 0 makes the observed three-bit pattern equal the contaminated vial's number; 000 identifies vial 0.
Source: OpenStax, College Algebra, Counting Principles https://openstax.org/books/college-algebra/pages/9-5-counting-principles
- 61
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.
Hint: The hen causes all the trouble. It goes first, and it may have to come back.
AnswerTake the hen across, return alone, take the fox across, bring the hen back, take the grain across, return alone, then take the hen across.
After the first trip, the fox and grain are safe together. Swapping the fox for the hen leaves the fox with grain across the river. Moving the grain next leaves the hen alone, and the final trip reunites all three without ever leaving a forbidden pair unattended.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-develop-key-problem-solving-skills
- 62
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?
Hint: A locker is toggled once for each number that divides it. Which numbers have an odd count of divisors?
AnswerThe ten lockers numbered by perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
Locker n is toggled once for each positive divisor of n. Divisors normally pair as d and n/d, giving an even number of toggles. Only perfect squares have one unpaired divisor, their square root, so exactly the square-numbered lockers receive an odd number of toggles and finish open.
Source: OpenStax, College Algebra, Counting Principles https://openstax.org/books/college-algebra/pages/9-5-counting-principles
- 63
Build a five-move word ladder from HEAD to TAIL, changing one letter per move.
Hint: Change the D of HEAD first, to make a word meaning to get better.
AnswerHEAD, HEAL, TEAL, TELL, TALL, TAIL.
Each adjacent pair changes exactly one letter: D/L, H/T, A/L, E/A, then L/I.
Source: Merriam-Webster.com Dictionary, "head" https://www.merriam-webster.com/dictionary/head
- 64
Pages are numbered from 1 through 120. How many times is the digit 1 printed?
Hint: Count the 1s in the ones place, the tens place and the hundreds place separately.
Answer53 times.
From 1 to 99 there are 20. From 100 to 120 there are 21 hundreds digits, 10 tens digits, and 2 ones digits: 20 + 21 + 10 + 2 = 53.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-develop-key-problem-solving-skills
- 65
A rectangle has positive whole-number side lengths and area 36. Which side lengths give the smallest perimeter?
Hint: The closer the two sides are to each other, the shorter the way around.
Answer6 by 6, with perimeter 24.
The factor pairs are 1x36, 2x18, 3x12, 4x9, and 6x6, with perimeters 74, 40, 30, 26, and 24.
Source: OpenStax, Prealgebra 2e, Rectangles, Triangles, and Trapezoids https://openstax.org/books/prealgebra-2e/pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids
- 66
A rectangle has positive whole-number side lengths and perimeter 22. Which dimensions give the greatest area?
Hint: Length plus width is 11. Keep the two sides as close as you can.
Answer5 by 6, with area 30.
The side lengths must total 11. The possible pairs 1x10, 2x9, 3x8, 4x7, and 5x6 have areas 10, 18, 24, 28, and 30, so 5 by 6 is greatest.
Source: OpenStax, Prealgebra 2e, Rectangles, Triangles, and Trapezoids https://openstax.org/books/prealgebra-2e/pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids
- 67
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.
Hint: 738 rules out 7, 3 and 8. Then 780 tells you 0 is in the code, but not last.
Answer042.
From 738, digits 7, 3, and 8 are absent. The 780 clue then makes 0 present but not third. In 682, 2 must be the well-placed third digit or 6 must be first; the 614 misplaced clue rules out a well-placed 6, so 6 is absent and 2 is third. The 206 clue puts 0 and 2 in the code but not in positions 2 and 1, so 0 is first. In 614, 1 cannot occupy its shown second position, leaving 4 as the one correct misplaced digit in the remaining second slot.
Source: NRICH, Using NRICH Tasks to Develop Key Problem-Solving Skills https://nrich.maths.org/articles/using-nrich-tasks-develop-key-problem-solving-skills
- 68
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?
Hint: Ana's and Bo's statements cannot both be false, so one of them is the true one.
AnswerBo.
If Ana took it, Ana's statement is false while Bo's and Cy's are true, giving two truths. If Cy took it, Ana's and Cy's are true, again giving two. If Bo took it, only Ana's statement is true: Bo's accusation is false and Cy's claim is false. Therefore Bo is the sole consistent culprit.
Source: Open Logic Project, forall x: Calgary https://forallx.openlogicproject.org/html/Ch11.html
How to play
Read the prompt once, give everyone quiet working time, then ask for the route before revealing the checked answer.
Timer and team scores
8 turns × 60 seconds = 8 minutes of play, plus instructions and breaks.
60
Pause whenever the group needs more time.
Team Lime's turn
Team Lime: 0
Team Violet: 0
Team Tangerine: 0
Team Sky: 0
Team Lemon: 0
Team Bubblegum: 0