I’ve posted puzzles before because they can teach us about how the human brain works. Spoiler: it works imperfectly.
Part 1: Counterintuitive Puzzles that Should Be Easy
Part 2: 12 More Puzzles
You may remember this one:
A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?
An answer will probably pop to mind—10 cents—but that’s wrong. (This and related puzzles were discussed in part 1.) Whether you give the intuitive answer or analyze further to find the correct answer says something about how you think. This is explored in an article provocatively titled, “3-Question Quiz Predicts Whether You Believe in God.”
Here are some more puzzles, just for fun (hints and answers are at the bottom).
Quick puzzles
1. “Jack is looking at Anne, and Anne is looking at George; Jack is married, George is not. Is a married person looking at an unmarried person?” Answers: (a) Yes, (b) No, (c) Not enough information. Source
2. Start with 100, then add 10% and then subtract 10%. How much do you have now?
3. Cup 1 holds milk, and cup 2 has an equal amount of coffee. Take a spoonful of milk from cup 1 and pour it into cup 2. Now take a spoonful from cup 2 and pour it into cup 1. Which cup is now more concentrated? Answers: (a) Cup 1 has a higher fraction of milk than cup 2 has coffee, (b) cup 2 has a higher fraction of coffee than cup 1 has milk, (c) the fractions are equal. Source
Puzzles about the real world
4. What happens to a helium balloon in a car when the car accelerates? Source
5. Why is a mirror left-right reversed? Why isn’t it top-down reversed, too?
6. Your canoe overturns, and you swim to shore. You’re looking for shelter, and it’s getting dark. Luckily, you find a cabin. Inside, there’s a box of matches and a kerosene lamp. While there is kerosene in the base of the lamp, the wick is too short to reach it. You’ve checked the cabin, and there’s no more kerosene or wicks. How do you light the lamp to signal for help?
Math puzzles
7. “Three people with different salaries need to find out their average salary without revealing individual salaries to each other. How?” Source
8. Three people rent a hotel room for $60, so they each pay $20. Later, the manager realized he charged too much—it was only supposed to be $55. He sends a porter to the room with $5. Each of the three guests keeps $1, and they give the porter the remaining $2 for a tip. Now each person has paid $19 for the room. But $19×3 = $57, and the porter got $2. They started with $60, and $57 + $2 = $59! Where is the other dollar? [I saw this one about 50 years ago.]
9. Joanne and her friends are seated around a large table. A plate with 25 cookies is passed around. Each person in turn takes a cookie and passes the plate along until it’s empty. The plate might go around once or several times, but the only rule is that the plate must start and end with Joanne (that is, she takes the first and last cookies). What are the possible values for the number of people sitting at the table? (h/t commenter Dave Gardner)
Hints and Answers are below.
Whatever Nature has in store for mankind,
unpleasant as it may be,
men must accept,
for ignorance is never better than knowledge.
— Enrico Fermi
Hints:
Problem 1: “Jack is looking at Anne, and Anne is looking at George; Jack is married, George is not. Is a married person looking at an unmarried person?” Answers: (a) Yes, (b) No, (c) Not enough information.
Hint: Okay, I’ll give you the answer: it’s (a). Now tell me why.
Problem 2: Start with 100, then add 10% and then subtract 10%. How much do you have now?
Hint: You’d think you’d return to 100, wouldn’t you?
Problem 3: Cup 1 holds milk, and cup 2 has an equal amount of coffee. Take a spoonful of milk from cup 1 and pour it into cup 2. Now take a spoonful from cup 2 and pour it into cup 1. Which cup is now more concentrated? Answers: (a) Cup 1 has a higher fraction of milk than cup 2 has coffee, (b) cup 2 has a higher fraction of coffee than cup 1 has milk, (c) the fractions are equal.
Hint: What if you used a really big spoon to make the transfer? How about a really small spoon? (And problem #2 is vaguely relevant.)
Problem 4: What happens to a helium balloon in a car when the car accelerates?
Hint: How does the balloon act compared to the air around it?
Problem 5: Why is a mirror left-right reversed? Why isn’t it top-down reversed, too?
Hint: Note that it’s not because you have two eyes—if you close one eye, the person in the mirror still raises the opposite hand that you do.
Problem 6: Your canoe overturns, and you swim to shore. You’re looking for shelter, and it’s getting dark. Luckily, you find a cabin. Inside, there’s a box of matches and a kerosene lamp. While there is kerosene in the base of the lamp, the wick is too short to reach it. You’ve checked the cabin, and there’s no more kerosene or wicks. How do you light the lamp to signal for help?
Hint: You don’t have anything to make the wick longer. You could tear your clothes to make material for a wick, but the clothes are wet. Really, really wet. They’re just full of water …
Problem 7: “Three people with different salaries need to find out their average salary without revealing individual salaries to each other. How?”
Hint: The answer involves a random number.
Problem 8: Three people rent a hotel room for $60, so they each pay $20. Later, the manager realized he charged too much—it was only supposed to be $55. He sends a porter to the room with $5. Each of the people keeps $1, and they give the porter the remaining $2 for a tip. So now each person has paid $19 for the room ($19×3 = $57) and the porter got $2. But they started with $60, and $57 + $2 = $59! Where is the other dollar?
Hint: It’s true that $60 ≠ $59, but is that really a problem?
Problem 9: Joanne and her friends are seated around a large table. A dish with 25 cookies is passed around. Each person in turn takes a cookie and passes the dish along until it’s empty. The dish might go around once or several times, but the only rule is that the dish must start and end with Joanne (that is, she takes the first and last cookies). What are the possible values for the number of people sitting at the table?
Hint: 24 people would work, because Joanne would take the first and 25th cookies, but 23 people wouldn’t work, since Joanne would take the next-to-last cookie but not the last. What’s the smallest number of people that would work? Also note that Joanne always gets one more cookie than everyone else.
Answers:
Problem 1: “Jack is looking at Anne, and Anne is looking at George; Jack is married, George is not. Is a married person looking at an unmarried person?” Answers: (a) Yes, (b) No, (c) Not enough information.
Answer: You’d think it would be (c), but it’s actually (a). If Anne is married, then she is a married person looking at an unmarried person. If Anne isn’t married, then Jack is a married person looking at an unmarried person. QED
Problem 2: Start with 100, then add 10% and then subtract 10%. How much do you have now?
Answer: 100 + 10% (10) = 110; then subtract 10% (11) to get 99.
Here’s another problem: Start again with 100. Now subtract 10% and then add 10% (close your eyes and figure it out before reading the answer).
Answer: This second computation is: 100 – 10% (10) = 90; then add 10% (9) to get 99.
But that raises yet another problem: why did the result get lower both times? Aren’t they symmetric? Doesn’t it seem like you should get something like 101 the second time?
Answer: You did the subtract-ten-percent bit on the bigger number both times.
Problem 3: Cup 1 holds milk, and cup 2 has an equal amount of coffee. Take a spoonful of milk from cup 1 and pour it into cup 2. Now take a spoonful from cup 2 and pour it into cup 1. Which cup is now more concentrated? Answers: (a) Cup 1 has a higher fraction of milk than cup 2 has coffee, (b) cup 2 has a higher fraction of coffee than cup 1 has milk, (c) the fractions are equal.
Answer: The answer is that they’re the same. What’s really crazy is that it doesn’t matter if you stir up the coffee with milk before you make the second transfer—the concentrations are the same either way.
Let’s explore a few different spoon sizes to see that this intuitively makes sense. Imagine that you start with one 8-ounce cup* of milk and another of coffee. Now imagine that the spoon is also one cup in volume (that is, you dump all the milk from cup 1 into the coffee and then take one cup of the mixture back to the empty cup 1). You have a 50/50 mixture in each cup but, more importantly for our puzzle, the amounts are equal. Now imagine that the spoon is a dropper. After the transfer, you’ll have cup 1 with 99+% milk and cup 2 with 99+% coffee. We didn’t do the math, but again, it looks like the amounts are the same. Now imagine that the spoon was so small that it had no effect—each cup would be at 100%, and again they’re equal.
Commenter MR suggests a discrete approach—have 16 white checkers in one pile and 16 black in another. Move n white checkers into the black pile, and then take n random ones from that pile back over. Say that n = 4, and try all five possibilities of the second transfer, from 4w + 0b (4 white and 0 black) to 0w + 4b. I find this a more intuitive approach.
Problem 4: What happens to a helium balloon in a car when the car accelerates?
Answer: A helium balloon acts the opposite of the air around it. When everything is pushed back in the car, the balloon goes forward. If the car turns left, you will feel pushed to the right … but the balloon is pushed left. Think of the vector of apparent gravity. When you’re motionless, it’s straight down, but when you’re accelerating, that vector changes to point more back. The balloon in air goes in the opposite direction of this vector.
Seen from a buoyancy standpoint, the air in the car is more affected by acceleration and gravity than the balloon is. The acceleration pushes the balloon back, too, but the air is pushed back more forcefully, pushing the balloon forward.
Problem 5: Why is a mirror left-right reversed? Why isn’t it top-down reversed, too?
Answer: Say you’re facing a wall with your right hand up and left down, and someone takes a picture of you from behind. Imagine a life-size version of that picture placed on the wall in front of you. That’s you from the back. It’s not reversed, since the image has its right hand up and left down, just like you.
Now imagine the camera was magic so that it captured, not the side facing it, but the side facing away. That is, it didn’t capture the appearance of the nearest atoms of you but the farthest atoms. Now put that image up in front of you, and it will look like a mirror. There’s no left-right reversal going on, it’s simply showing you the image of your front, seen from the back. It’s like you were transparent except for your front surface. That’s what a mirror shows you—a mirror image, not a reversal.
Problem 6: Your canoe overturns, and you swim to shore. You’re looking for shelter, and it’s getting dark. Luckily, you find a cabin. Inside, there’s a box of matches and a kerosene lamp. While there is kerosene in the base of the lamp, the wick is too short to reach it. You’ve checked the cabin, and there’s no more kerosene or wicks. How do you light the lamp to signal for help?
Answer: The kerosene floats on water, so pour enough water into the base of the lamp to raise the kerosene to reach the wick. You could put rocks in instead, but that wouldn’t be as efficient.
Problem 7: “Three people with different salaries need to find out their average salary without revealing individual salaries to each other. How?”
Answer: The first person adds a random number to their salary and secretly passes that to the second person, who adds their correct salary and secretly passes the sum to #3, who does the same. Then #1 subtracts the random number and divides by 3 to get the average salary.
Problem 8: Three people rent a hotel room for $60, so they each pay $20. Later, the manager realized he charged too much—it was only supposed to be $55. He sends a porter to the room with $5. Each of the people keeps $1, and they give the porter the remaining $2 for a tip. So now each person has paid $19 for the room ($19×3 = $57) and the porter got $2. But they started with $60, and $57 + $2 = $59! Where is the other dollar?
Answer: It’s meaningless to compute $57 + $2 = $59 because that mixes the before room rate ($60) with the money paid after the room rate was corrected (that is, after each guest got $1 and the porter got $2). The room’s cost afterwards was $55, so the computation should be $57 – $2 = $55. A different problem is, “Hey—what happened to the $60?” To answer that, you start with the $55 in the manager’s possession and then return the dollar refund from each guest and the tip from the porter ($1×3 + $2 = $5).
Problem 9: Joanne and her friends are seated around a large table. A dish with 25 cookies is passed around. Each person in turn takes a cookie and passes the dish along until it’s empty. The dish might go around once or several times, but the only rule is that the dish must start and end with Joanne (that is, she takes the first and last cookies). What are the possible values for the number of people sitting at the table?
Answer: We need to figure out how many complete turns of the table will work by finding integer solutions to nm + 1 = 25, where n is the number of friends at the table and m is the number of turns. For example, for the 24-person case, n = 24 and m = 1. Add 1 because lucky Joanne always gets one more to get one answer: n + 1 = 25 total people at the table.
So what integer values of n and m will work? We know that m = 1 works, and that gives n = 24. Other (m, n) pairs that work are (2, 12), (3, 8), (4, 6), (6, 4), (8, 3), and (12, 2). But our equation doesn’t have integer values for n if m is any of the remaining values: 5, 7, 9, 10, or 11.
Remember that n is the other people sitting at the table, so the answer to “how many people are at the table, in total (including Joanne)?” is n + 1. That means that the possible values for the number of people at the table are 25, 13, 9, 7, 5, 4, and 3.
*Yes, I’m using God’s own units, and I won’t apologize for that. If Liberia and Myanmar don’t need the metric system as their official system of weights and measures, then neither does Uncle Sam.