13 Math Puzzles with Elegant Answers STAYORSWITCH.COM The following math brainteasers are a collection of problems I’ve heard from friends and found on the internet. I selected them because they are challenging, but also easy to understand with elegant solutions. I use them in my math classes. They are in no particular order and none of them have “trick” answers. Enjoy! 1. 40 Pounds of Math. A miller uses a 40 lb rock to measure grain with a two-­‐sided scale. He lends the rock to his friend the butcher. The butcher comes to return the rock and says, “I’m sorry but I dropped the rock and it broke into 4 pieces.” Upon seeing the pieces, the miller says, “Actually, this is great. With these 4 pieces I can measure any amount of grain from 1 to 40 lbs!” What are the weights of the 4 pieces? Note: using a two-­‐sided scale, if you have a 6 lb and a 10 lb rock, you can measure 4 lbs. 2. Crazy Train. 100 people are lined up to get on a train with 100 seats. Each person has a ticket for their assigned seat and they’re lined up from 1-­‐100. The first person in line has a ticket for seat #1, the second person’s ticket is seat #2, etc., all the way to the 100th person. However, the first person in line is a “crazy” person and instead of taking his assigned seat, he randomly chooses one of the 100 seats on the train. Everyone after him goes to their assigned seat, unless their assigned seat is taken in which case they become the new “crazy” person and pick a random seat out of the remaining available seats. What is the probability that the 100th person gets his assigned seat? 3. Can Someone Crack a Window? There are 100 windows. The first person opens all of the windows. The second person closes every second window. The third person goes to every third window and closes it if it’s open or opens it if it’s closed. The fourth person goes to every fourth window and does the same as the third person. After the hundredth person, how many windows are open? 4. Spider. A spider is in the bottom left corner of a 10’ x 10’ x 10’ room. If the spider can only move along walls and not through open space, what is the shortest distance it needs to travel to the opposite corner (top right) of the room? 5. Magic Trick with Six Cards. In a magic trick, there are three participants: the magician, an assistant, and a volunteer. The magician, who claims to have paranormal abilities, goes into a soundproof room. The assistant gives the volunteer six blank cards, five white and one blue. The volunteer writes a different integer from 1 to 100 on each card, as the assistant is watching. The volunteer keeps the blue card. The assistant arranges the five white cards in some order and passes them to the magician. The magician then announces the number on the blue card. How does the trick work? 6. Horse Racing. There are 25 horses, only five can be raced at a time, and there is no way to time the horses. Find the 3 fastest horses in as few races as possible. 7. Burning Your Rope. There are 2 ropes and a box of matches. Each of the ropes takes an hour to burn from end to end. However, they don't burn uniformly, so one inch of the rope could take 50 minutes to burn through, and the rest could burn up in 10 minutes. Measure 15 minutes of time. 1 8.
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Buckets O’ Balls. At a job interview, your potential employer presents you with the following test: There are 2 buckets, and a bin with 50 green balls and 50 red balls. He tells you he will leave the room, and that you must place the balls in the buckets. When he comes back, he will randomly select a bucket (with equal probability), and randomly draw a ball from that bucket. If he draws a green ball, you are hired. No bucket can be empty and each of the 100 balls must be placed in one of the two buckets. How can you arrange the balls to maximize your chances of getting the job? It’s a Girl! James and Christine are a married couple. They have two kids; one of them is a girl. What is the probability that the other kid is also a girl? (Assume that the probability of each gender is ½.) 10. Guessing Numbers. I’m thinking of a number that is 1, 2, or 3. You can ask just 1 question about the number. I can only answer "Yes", "No", or "I don't know". After I answer your question, you know what the number is. What is the question? 11. Soda Recycling. The grocery store lets you exchange 3 empty soda cans for another soda. The soccer team has 17 players and each bought a can of soda after the game. How many more cans of soda can the soccer team get by exchanging? 12. Be There or Be Square. = 5 = ? 13. Dragonfly. Two boats 200 miles apart are moving toward each other; each one is going at a speed of 50 miles per hour. A dragonfly starting on the front of one of them flies back and forth between them at a rate of 75 miles per hour. It does this until the boats collide and crush the dragonfly to death. What is the total distance the dragonfly flew before the collision? I hope you enjoyed the math puzzles. Let me know what you think! Please send any feedback, comments, or suggestions to [email protected]. 2 ANSWERS 1. 40 Pounds of Math. The 4 pieces weigh 1, 3, 9, 27 (powers of 3). With these four weights, any weight can be measured up to 40 lbs. 1 lb can be measured with the 1 lb piece. 2 lbs can be measured with the 3 lb piece on one side and the 1 lb piece on the other side of the scale, along with whatever is being weighed. The powers of 3 have this property ongoing. The same question could be posed for an initial rock of 121 lbs (which seems a bit heavy) that breaks into 5 pieces (1, 3, 9, 27, 81). 2. Crazy Train. 50%. As each “crazy” person randomly selects their seat, we can categorize their choices into: • Chooses a seat from #2-­‐99 and delays the determination of the outcome • Chooses seat #1 and guarantees the 100th person will get seat #100 • Chooses seat #100 and guarantees the 100th person will NOT get seat #100 Because for each new “crazy” person there is always an equal probability of seat #1 or #100 being chosen, the chance that the 100th person gets their seat is 50%. If we were to change the second-­‐to-­‐last sentence of the problem, it becomes easier to think about. “Everyone after him goes to their assigned seat. If the “crazy” person is sitting there, then the “crazy” person must get up and pick another random seat out of the remaining available seats.” In this situation, there’s only ever one person who is out of their assigned seat, just getting bumped around until they eventually choose seat #1 or #100. 3. Can Someone Crack a Window? 10 windows are left open. Each window gets toggled once for each of its factors. So the 12th window will get toggled 6 times because there are 6 factors of 12 (1, 2, 3, 4, 6, 12). The windows that are toggled an odd number of times will remain open at the end. The only numbers with an odd number of factors are perfect squares. That means there will be 10 windows left open at the end because there are 10 perfect squares from 1-­‐100 (1, 4, 9, 16, 25, 36, 49, 64, 81, 100). 4. Spider. 10 5 or about 22.4 feet. Because the spider must travel along the surfaces of the walls, this problem is most effectively solved using a net of the room (think of unfolding a paper cube). The room’s net would look like this: 10 10 10 The arrow represents one of the two shortest paths for the spider, from the bottom left to the top right corner of the room. This illustrates that the spider’s distance is the hypotenuse of a right triangle with legs measuring 10’ and 20’. So, ! = !2 + !2 = 102 + 202 = 500 = 10 5. 3 5. Magic Trick with Six Cards. Any five different numbers can be ordered in 120 different ways (5 factorial, permutation of 5 choose 5). Therefore, with an agreed-­‐upon system, the assistant can arrange the five white cards in a specific order to indicate the number on the blue card. For example, let’s say the numbers that the volunteer wrote on the white cards are 9, 23, 54, 61, 80, and the volunteer wrote 50 on the blue card. If you organize the numbers on the white cards from least to greatest, you can think of them as 1, 2, 3, 4, 5. Using one system, if the cards were organized as 1, 2, 3, 4, 5 (9, 23, 54, 61, 80), it would indicate the number 1. If they were organized as 5, 4, 3, 2, 1 (80, 61, 54, 23, 9), it would indicate the number 120. So the number 50 would be communicated with an ordering of 3, 1, 2, 5, 4 (54, 9, 23, 80, 61). Here’s how: • 3, being the third lowest number, indicates that it's in the third set of 24 (49-­‐72) • 1, being the first lowest remaining number, indicates that it's in the first set of 6 within that set of 24 (49-­‐55) • 2, being the first lowest remaining number indicates that it's in the first set of 2 within that set of 6 (49-­‐50) • 5, being the second lowest remaining number indicates that it's in the second set of 1 within that set of 2 (50) • The 4 at the end is inconsequential This system can be used to communicate any number from 1-­‐120. Here’s a more in-­‐depth explanation: Because there are five numbers to choose from for the first digit of the code (1-­‐5), we can break 120 into fifths and have the first number indicate which fifth the blue card number (BCN) is in. So if the code starts with 1, the BCN is in the first fifth of 120—from 1-­‐24. If the code starts with 2, it indicates that the BCN is in the second fifth of 120—from 25-­‐48. After the first digit is chosen, there are 4 numbers available for the second digit. Because there are 4 numbers to choose from for the second digit of the code, we can break the 24-­‐number range into fourths and have the second digit indicate which fourth the BCN is in. For example, if the first digit was 1 (indicating that the BCN is somewhere from 1-­‐24), and the second digit was 3, the 3 indicates that the BCN is in the second fourth of 24—from 7-­‐12. The 3 indicates the second fourth because it is the second lowest of the remaining numbers (2, 3, 4, 5). After the first two digits are chosen, there are 3 numbers available for the third digit. Because there are 3 numbers to choose from for the third digit of the code, we can break the 6-­‐number range into thirds and have the third digit indicate which third the BCN is in. Continuing our example, if the third digit was 5, the 5 would indicate that the BCN is in the third third of 7-­‐12—
from 11-­‐12. After the first three digits are chosen, there are only 2 numbers to choose from for the fourth digit. The fourth digit simply indicates if the BCN is the first or second of the two-­‐number range. If the digit is the smaller of the two remaining numbers, then it’s the first in the range. If it’s the larger of the two, it indicates the second number in the range. So if the fourth digit is a 2, the 2 indicates that the BCN is the first number in the range of 11-­‐12—11. So a code of 13524 communicates a blue card number of 11. Easy! 4 6. Horse Racing. It takes 7 races. Let’s say there are 5 groups numbered 1, 2, 3, 4, 5 and within each group there are five horses labeled A, B, C, D, E. Start by racing each group, that's 5 races. In each numbered group, we can assign A to the fastest horse and E to the slowest horse. After the first five races, race the winners from each group (1A, 2A, 3A, 4A, 5A) against each other. This is the sixth race. Let's say the horse from group 1 was the fastest all the way down to the horse from group 5 who was the slowest in that race. So now we know that 1A is the fastest. This leaves us with four possible top-­‐three rankings: • 1A 1B 1C • 1A 1B 2A • 1A 2A 2B • 1A 2A 3A This means there are only 5 horses left to compete for the 2nd and 3rd positions: 1B, 1C, 2A, 2B, 3A. You race them in the 7th race. 7. Burning Your Rope. Light both ends of one rope and one end of the other rope. After 30 minutes, the first rope will burn out completely. The second rope has burned for 30 minutes with 30 minutes remaining. Light the other end of the second rope and it will take 15 minutes from that point until it burns out completely. 8. Buckets O’ Balls. Place one green ball in one of the buckets and the rest of the balls (49 green, 50 red) in the other bucket. To calculate the exact probability of getting the job, we can find the probability of picking a green ball from each bucket, and then average the two probabilities. The chance of picking a green ball from the bucket with one green ball is 100%. The chance of picking a green ball from the other bucket is about 49.5% (49 green balls out of 99 total balls). Average those two percentages and your probability of getting the job is about 75%. 9. It’s a Girl! There is a 1 3 chance that the other child is also a girl. The following are possible scenarios of two children in a family: Girl -­‐ Girl Girl -­‐ Boy Boy -­‐ Girl Boy -­‐ Boy Since we know one of the children is a girl, it eliminates the Boy-­‐Boy possibility from the options. This leaves only three possibilities, one of which is two girls, making the probability that the other child is a girl 1 3. 10. Guessing Numbers. "I'm also thinking of a number. It's either 1 or 2. Is my number less than yours?" There are other solutions as well. 11. Soda Recycling. 8 new sodas. 15 of the 17 cans can be exchanged for 5 sodas, with 2 extra cans. After drinking the 5 new sodas, there are 7 cans. Exchange 6 of the 7 cans for 2 more sodas, with 1 extra can. After drinking the 2 new sodas, there are 3 cans remaining. Exchange the final three cans for 1 more soda. So the soccer team was able to get another 8 (5 + 2 +1) cans of soda. 5 12. Be There or Be Square. 14. The number associated with the square is the number of squares that can be seen in the diagram. The first square has 5 squares, the 4 smaller ones, along with the 2×2 square. Counting 1×1, 2×2, and 3×3 squares, the second picture has 14 squares. With this reasoning, a 4×4 grid would have 30 squares. In general, an n×n grid would have 12 + 22 + … + n2 squares. The resulting numbers (1, 5, 14, 30, etc.) are called the square pyramidal numbers, because they are the number of cubes needed to build a pyramid with an n×n square base that is n levels high. 13. Dragonfly. 150 miles. The dragonfly actually touches each boat an infinite number of times before it gets crushed, and one could solve the problem the hard way with pencil and paper by summing an infinite series of distances. But there’s an easier way. Since the boats are 200 miles apart and each one is going 50 miles an hour, it takes 2 hours for the boats to collide. Therefore the dragonfly was flying for two hours. Since it was flying at a rate of 75 miles per hour, it must have flown 150 miles. 6