Answers:
1. If fence posts are put in every 7 feet, how many posts are needed to make a 77-foot fence? 12 posts. If you said 11, you made a “fencepost error,” named after this puzzle, for forgetting that there’s a post on each end. A 7-foot-long fence would have 2 posts. Date spans are similarly tricky: “I’ll be away on vacation from August 5 through August 23.” How many days is that? Not 23 – 5 = 18 but rather 19.
2. If it takes a chiming clock 3 seconds to strike 6:00, how long to strike midnight? When striking 6:00, you have 5 gaps between chimes that, in total, span 3 seconds. That makes 3/5 seconds per gap. There are 11 gaps when striking midnight, so that’s 3/5 seconds/gap × 11 gaps = 33/5 = 6.6 seconds.
3. Sentence sense. These are called garden-path sentences.
- The old man the boat: Change a few synonyms to get: Old people operate the boat.
- While the man hunted, the deer ran into the woods.
- I convinced her that children are noisy
- The coach smiled at the player who was tossed the Frisbee.
- The cotton that clothes are made up of grows in Mississippi.
- The horse that was raced past the barn fell.
4. Where does the length of a year come from? One year is the time it takes the earth to go around the sun. It has no connection with the definition of a day, which is why there’s an arbitrary number of days in a year (365.242199…).
5. Why is it colder in the winter? The spinning earth isn’t vertical to the plane of the earth’s orbit but tilts about 23°. The winter solstice (around December 21 in the northern hemisphere) is the shortest day of the year because it’s the day when your hemisphere’s pole points farthest away from the sun. That means less sun, and less sun means colder.
6. Rowboat drops cannonball into the swimming pool.
We deal with water and buoyancy all the time—we displace water when we step into a bath, we plunge our hands into a sink full of water, we float on an inner tube. And yet this simple puzzle stumps many of us.
The relevant properties of the cannonball are its weight and its volume. Imagine the rowboat in the pool. Once you add the 20kg cannonball to the rowboat, the rowboat is now 20kg heavier and so displaces 20kg more water. Only the weight of the cannonball is interesting at this stage—20kg of feathers would have the same effect. The water level on the side of the pool rises the same amount as if 20kg of water had been poured in.
Now drop the cannonball overboard. The rowboat is lighter—it rides higher in the water, and the water level of the pool goes down. But what about the cannonball in the water? Now the cannonball’s volume is all that matters. The tiny 20kg pellet we imagined would have basically zero volume and so wouldn’t change the water level noticeably if you tossed it out of the rowboat. The answer to the puzzle is that the water level in the pool goes down.
The cannonball doesn’t have zero volume, but it has a lot less volume than 20kg of water (steel is 8 times denser than water; lead even more so). While the cannonball’s volume would push up the water level a bit, it would be far less than its impact on the water level when in the boat. The answer is the same: the water level goes down.
If you like learning about the physics of everyday things—what makes rainbows, why the tea leaves in a cup go in when you stir instead of being flung to the outside, why spinning tops are stable, and more—I suggest The Flying Circus of Physics by Jearl Walker.
7. The average IQs of Iowa and Missouri both rise, but both states can’t improve, right? Both states can indeed improve. There is no Law of Conservation of Averages like there can be for energy or momentum. New Iowa now has fewer people and New Missouri now has more, so the averages have been recomputed. (h/t commenter Michael Neville)
This has been called the Will Rogers phenomenon. He reportedly claimed a similar intelligence transfer when migrants from Oklahoma moved to California during the Depression.
8. Proof that 1 = 2. The clue is in the first step: a = b. Remember the step “Cancel (a – b) from both sides.” If a = b, then (a – b) = 0. “Cancel (a – b)” is then dividing by zero. That’s a no-no, and now you see why.
9. Fire on the island. Use your matches to start a fire right where you are. The wind will blow it to the south, the direction you want to go to get away from the fire. By the time the fire nears your location, you will have a large burned-up area in front of you. Without fuel, you can stand in it and be safe from the oncoming wildfire.
10. Coffee with milk on the side. Let’s suppose that the air temperature is 75°. The coffee isn’t boiling, but it’s pretty close: 200°. The milk is at refrigerator temperature, say 35°. The coffee is much hotter than ambient than the milk is colder: 130° vs. 40°. Over time, the milk will get warmer (good), but the coffee will get colder (bad). Since the temperature difference for the coffee is much greater, it will be losing heat much faster than the milk will be gaining it, so that 15 minutes of wait will hurt the coffee’s temperature more than it will help the milk’s. Solution: pour the milk in now to lower the coffee’s temperature so that the mixture will lose heat more slowly.
11. Rolling dice. These dice are interesting because they are nontransitive. The transitive property says for some relation |, if A | B and B | C, then A | C. Example: if A > B and B > C, then A > C. These dice do not have that property.
Here’s one example of a cycle of four dice with that property:
- A: 4, 4, 4, 4, 0, 0
- B: 3, 3, 3, 3, 3, 3
- C: 6, 6, 2, 2, 2, 2
- D: 5, 5, 5, 1, 1, 1
Each die has a 2/3 chance of beating the next one down in the cycle—A beats B, which beats C, which beats D, which beats A. This means that if you let your opponent pick a die first, you can pick the next one up in the cycle to have a 2/3 chance of winning any roll in a contest.
The same nontransitive relation can be seen in four overlapping rings here. Each ring is completely over (not interlocked with) its next neighbor, and yet the entire thing is interlocked. The Borromean rings (three rings interlocked in the same overlapping way) have been used to represent the Trinity.
12. Does the balance tip right, left, or not at all? You can figure out the up and down forces on each side of the beaker, but that’s the hard way. Let’s simplify instead.
The left side can be simplified by cutting the string (the string does nothing in the same way that your weight doesn’t change when you stand on a scale and pull up on your shoelaces). Take the ping pong ball out of the left beaker and put it on the left side of the scale.
Now we get to the tricky part, the right side. You might think that since the arm holding the lead ball (I imagine the gray ball made out of lead, but it can be anything denser than water) is standing on the table, not the scale, the ball has no influence. This would be wrong. To see this, imagine a beach ball in a swimming pool. If you want to push it under water, you must push down with considerable force. If it’s a big ball, it will support all your weight. This is because pushing the beach ball down into the water means that you are displacing water. Displacing water means that you are pushing up water to get it out of the way. Water is heavy. A beach-ball sized pile of water is a lot of water to displace.
Now imagine that the lead ball suspended by a string were instead a ping pong ball pushed down by a rod. If you are pushing down, you’re pushing down into the beaker. The beaker gets heavier.
Here’s an alternative way to see the same downward force. Suppose the lead ball weighs 100g, and you have it suspended on a spring scale. The scale reads 100g. Now you lower the ball into the water. Obviously the ball is heavier than water and it won’t float, but it is supported somewhat by the water. That “somewhat” will register as the scale reading a little less—about 10g less. Where did that 10g go? It made the right side of the scale that much heavier. The 10g is how much a ball of water that size would weigh, since the lead ball displaced that much water.
Conclusion: the weight of the left side of the balance is equivalent to one beaker full of water + one ping pong ball. The right side is one beaker full of water + one water ball. Water ball > ping pong ball, so the scale tips to the right.
Do you have any good puzzles? Add them to the comments!