#### 1. THE POLE-CLIMBING SLOTH

A slippery sloth climbs six feet up a utility pole during the day, then slides back down five feet during the night. If the pole is 30 feet high and the sloth starts from the ground (zero feet), how many days does it take the sloth to reach the top of the pole?

**Answer**: 25 days. The math here boils down to a net gain of one foot per day, along with a threshold (24 feet at the beginning of a day) that must be attained so that the sloth can get to the 30-foot mark within a given day. After 24 days and 24 nights, the sloth is 24 feet up. On that 25th day, the sloth scrambles up six feet, attaining the 30-foot top of the pole. Left to the reader is a motivation for the sloth to attempt this feat in the first place. Perhaps there is something tasty atop the pole?

*(Adapted from a brain teaser by Carl Proujan.)*

#### 2. THE PIRATE RIDDLE

A group of five pirates have to divide up their bounty of 100 coins, as described in the video below. The captain gets to propose a distribution plan, and all five of the pirates vote "yarr" or "nay" on the proposal. If a majority votes "nay," the captain walks the plank. The pirates are arranged in order, and vote in that order: the captain, Bart, Charlotte, Daniel, and Eliza. If a majority vote "nay" and the captain walks the plank, the captain's hat goes to Bart, and the process repeats down the line, with a series of proposals, votes, and other acceptance or plank-walking.

How can the captain stay alive, while getting as much gold as possible? (In other words, what is the optimal amount of gold the captain should offer to each pirate, himself included, in his proposal?) Watch the video below for all the rules.

**Answer**: The captain should propose that he keep 98 coins, distribute one coin each to Charlotte and Eliza, and offer nothing to Bart and Daniel. Bart and Daniel will vote nay, but Charlotte and Eliza have done the math and vote yarr, knowing that the alternative would get them even less booty.

#### 3. THE HIKER'S DILEMMA

A hiker comes across an intersection where three roads cross. He looks for the sign indicating the direction to his destination city. He finds that the pole carrying three city names and arrows pointing to them has fallen. He picks it up, considers it, and pops it back into place, pointing out the correct direction for his destination. How did he do it?

**Answer**: He knew which city he had just come from. He pointed that arrow back toward his origin point, which oriented the signs properly for his destination and a third city.

*(Adapted from a brain teaser by Jan Weaver.)*

#### 4. THE PASSCODE RIDDLE

In the video below, the rules of this riddle are laid out. Here's a snippet: Three team members are imprisoned, and one is allowed the opportunity to escape by facing a challenge. Given perfect logical skills, how can the remaining two team members listen in on what the chosen team member does, and infer the three-digit passcode to get them out?

**Answer**: The passcode is 2-2-9, for hallway 13.

#### 5. COUNTING BILLS

I had a wad of money in my pocket. I gave half away and of what remained, I spent half. Then, I lost five dollars. That left me with just five bucks. How much money did I start with?

**Answer**: 40 dollars.

*(Adapted from a brain teaser by Charles Booth-Jones.)*

#### 6. THE AIRPLANE FUEL RIDDLE

Professor Fukanō plans to circumnavigate the world in his new airplane, as shown in the video below. But the plane's fuel tank doesn't hold enough for the trip—in fact, it holds only enough for half the trip. Fukanō has two identical support planes, piloted by his assistants Fugori and Orokana. The planes can transfer fuel in midair, and they must all take off from and land at the same airport on the equator.

How can the three cooperate and share fuel so that Fukanō gets all the way around the world and nobody crashes? (Check the video for more details.)

**Answer**: All three planes took off at noon, flying west, fully loaded with fuel (180 kiloliters each). At 12:45, each plane has 135 kl remaining. Orokana gives 45 kl to each of the other two planes, then heads back to the airport. At 14:15, Fugori gives another 45 kl to the professor, then heads back to the airport. At 15:00, Orokana flies *east*, effectively flying *toward* the professor around the globe. At precisely 16:30, Orokana gives him 45 kl and flips around, now flying alongside the professor. Meanwhile, Fugori takes off and heads for the pair. He meets them at 17:15 and transfers 45 kl to each plane. All three planes now have 45 kl and make it back to the airport.

#### 7. THE HAYSTACK PROBLEM

A farmer has a field with six haystacks in one corner, a third as many in another corner, twice as many in a third corner, and five in the fourth corner. While piling the hay together in the center of the field, the farmer let one of the stacks get scattered all over the field by the wind. How many haystacks did the farmer end up with?

**Answer**: Just one. The farmer had piled them all up the middle, remember?

*(Adapted from a brain teaser by Jan Weaver.)*

#### 8. THE THREE ALIENS RIDDLE

In this video riddle, you have crashed landed on a planet with three alien overlords named Tee, Eff, and Arr. There are also three artifacts on the planet, each matching a single alien. To appease the aliens, you need to match up the artifacts with the aliens—but you don't know which alien is which.

You are allowed to ask three yes-or-no questions, each addressed to any one alien. You can choose to ask the same alien multiple questions, but you don't have to.

It gets more complex, though, and this wickedly tricky riddle is best explained (both its problem and its solution) by watching the video above.

#### 9. THE FARMER'S WILL

One day, a farmer decided to do some estate planning. He sought to apportion his farmland among his three daughters. He had twin daughters, as well as a younger daughter. His land formed a 9-acre square. He wanted the eldest daughters to get equally sized pieces of land, and the younger daughter to get a smaller piece. How can he divide up the land to accomplish this goal?

**Answer**: Shown above are three possible solutions. In each, the box marked 1 is a perfect square for one twin, and the two sections marked 2 combine to make a square of the same size for the second twin. The area marked 3 is a small perfect square for the youngest child.

*(Adapted from a brain teaser by Jan Weaver.)*

#### 10. COINS

In my hand I have two American coins that are currently minted. Together, they total 55 cents. One isn't a nickel. What are the coins?

**Answer**: A nickel and a 50-cent piece. (Lately the U.S. 50-cent piece features John F. Kennedy.)

*(Adapted from a brain teaser by Jan Weaver.)*

#### 11. THE BRIDGE RIDDLE

A student, a lab assistant, a janitor, and an old man need to cross a bridge to avoid being eaten by zombies, as shown in the video below. The student can cross the bridge in one minute, the lab assistant takes two minutes, the janitor takes five minutes, and the professor takes 10 minutes. The group only has one lantern, which needs to be carried on any trip across. The zombies arrive in 17 minutes, and the bridge can only hold two people at a time. How can you get across in the time allotted, so you can cut the rope bridge and prevent the zombies from stepping on the bridge and/or eating your brains? (See the video for more details!)

**Answer**: The student and lab assistant go together first, and the student returns, putting three minutes total on the clock. Then, the professor and the janitor take the lantern and cross together, taking 10 minutes, putting the total clock at 13 minutes. The lab assistant grabs the lantern, crosses in two minutes, then the student and lab assistant cross together just in the nick of time—a total of 17 minutes.

#### 12. LITTLE NANCY ETTICOAT

Here's a nursery rhyme riddle:

Little Nancy Etticoat

In her white petticoat

With a red nose—

The longer she stands

The shorter she grows

Given this rhyme, what is "she?"

**Answer**: A candle.

*(Adapted from a brain teaser by J. Michael Shannon.)*

#### 13. THE GREEN-EYED LOGIC PUZZLE

In the green-eyed logic puzzle, there is an island of 100 perfectly logical prisoners who have green eyes—but they don't know that. They have been trapped on the island since birth, have never seen a mirror, and have never discussed their eye color.

On the island, green-eyed people are allowed to leave, but only if they go alone, at night, to a guard booth, where the guard will examine eye color and either let the person go (green eyes) or throw them in the volcano (non-green eyes). The people don't know their own eye color; they can never discuss or learn their own eye color; they can only leave at night; and they are given only a single hint when someone from the outside visits the island. That's a tough life!

One day, a visitor comes to the island. The visitor tells the prisoners: "At least one of you has green eyes." On the 100th morning after, all the prisoners are gone, all having asked to leave on the night before. How did they figure it out?

Watch the video for a visual explanation of the puzzle and its solution.

**Answer**: Each person can't be sure whether they have green eyes. They can only deduce this fact by observing the behavior of the other members of the group. If each person looks at the group and sees 99 others with green eyes, then logically speaking, they must wait 100 nights to give the others opportunities to stay or leave (and for each to make that calculation independently). By the 100th night, using inductive reasoning, the entire group has offered every person in the group an opportunity to leave, and can figure that it's safe to go.

#### 14. THE NUMBER ROW

The numbers one through 10, below, are listed in an order. What is the rule that causes them to be in this order?

8 5 4 9 1 7 6 10 3 2

**Answer**: The numbers are ordered alphabetically, based on their English spelling: eight, five, four, nine, one, seven, six, ten, three, two.

*(Adapted from a brain teaser by Carl Proujan.)*

#### 15. THE COUNTERFEIT COIN PUZZLE

In the video below, you must find a single counterfeit coin among a dozen candidates. You're allowed the use of a marker (to make notes on the coins, which doesn't change their weight), and just three uses of a balance scale. How can you find the one counterfeit—which is slightly lighter or heavier than the legitimate coins—among the set?

**Answer**: First, divide the coins into three equal piles of four. Put one pile on each side of the balance scale. If the sides balance (let's call this Case 1), all eight of those coins are real and the fake must be in the other pile of four. Mark the legitimate coins with a zero (circle) using your marker, take three of them, and weigh against three of the remaining unmarked coins. If they balance, the remaining unmarked coin is counterfeit. If they don't, make a different mark (the video above suggests a plus sign for heavier, minus for lighter) on the three new coins on the scale. Test two of these coins on the scale (one on each side)—if they have plus marks, the heavier of those tested will be the fake. If they have minus marks, the lighter is the fake. (If they balance, the coin not tested is the fake.) For Case 2, check out the video.

#### 16. THE ESCALATOR RUNNER

Each step of an escalator is 8 inches taller than the previous step. The total vertical height of the escalator is 20 feet. The escalator moves upward one half step per second. If I step on the lowest step at the moment it is level with the lower floor, and run up at a rate of one step per second, how many steps do I take to reach the upper floor? (Note: Do not include the steps taken to step on and off the escalator.)

**Answer**: 20 steps. To understand the math, take a period of two seconds. Within that two seconds, I run up two steps on my own power, and the escalator lifts me the height of an extra step, for a total of three steps—this could also be expressed as 3 times 8 inches, or two feet. Therefore, over 20 seconds I reach the upper floor having taken 20 steps.

*(Adapted from a brain teaser by Carl Proujan.)*

#### 17. A RIVER CROSSING PUZZLE

In the video riddle below, three lions and three wildebeest are stranded on the east bank of a river and need to reach the west. A raft is available, which can carry a maximum of two animals at a time and needs at least one animal onboard to row it across. If the lions ever outnumber the wildebeest on either side of the river (including the animals in the boat if it's on that side), the lions will eat the wildebeest.

Given these rules, how can all the animals make the crossing and survive?

**Answer**: There are two optimal solutions. Let's take one solution first. In the first crossing, one of each animal goes from east to west. In the second crossing, one wildebeest returns from west to east. Then on the third crossing, two lions cross from east to west. One lion returns (west to east). On crossing five, two wildebeest cross from east to west. On crossing six, one lion and one wildebeest return from west to east. On crossing seven, two wildebeest go from east to west. Now all three wildebeest are on the west bank, and the sole lion on the west bank rafts back to the east. From there (crossings eight through eleven), lions simply ferry back and forth, until all the animals make it.

For the other solution, consult the video.

#### 18. THE THREE WATCHES

I am marooned on an island with three watches, all of which were set to the correct time before I got stuck here. One watch is broken and doesn't run at all. One runs slow, losing one minute every day. The final watch runs fast, gaining one minute every day.

After being marooned for a moment, I begin to worry about timekeeping. Which watch is most likely to show the *correct time* if I glance at the watches at any particular moment? Which would be *least* likely to show the correct time?

**Answer**: We know that the stopped watch must tell the correct time twice a day—every 12 hours. The watch that loses one minute per day will not show the correct time until 720 days into its cycle of time loss (60 minutes in an hour times 12 hours), when it will momentarily be exactly 12 hours behind schedule. Similarly, the watch that gains one minute a day is also wrong until 720 days after its journey into incorrectness, when it will be 12 hours ahead of schedule. Because of this, the watch that doesn't run at all is most likely to show the correct time. The other two are equally likely to be incorrect.

*(Adapted from a brain teaser by Carl Proujan.)*

#### 19. EINSTEIN'S RIDDLE

In this riddle, erroneously attributed to Albert Einstein, you're presented with a series of facts and must deduce one fact that's not presented. In the case of the video below, a fish has been kidnapped. There are five identical-looking houses in a row (numbered one through five), and one of them contains the fish.

Watch the video for the various bits of information about the occupants of each house, the rules for deducing new information, and figure out where that fish is hiding! (Note: You really need to watch the video to understand this one, and the list of clues is helpful too.)

**Answer**: The fish is in House 4, where the German lives.

#### 20. MONKEY MATH

Three castaways and a monkey are marooned together on a tropical island. They spend a day collecting a large pile of bananas, numbering between 50 and 100. The castaways agree that the next morning the three of them will divide up the bananas equally among them.

During the night, one of the castaways wakes up. He fears that the others might cheat him, so he takes his one-third share and hides it. Since there is one banana more than a quantity which could be divided equally into thirds, he gives the extra banana to the monkey and goes back to sleep.

Later in the night, a second castaway awakes and repeats the same behavior, plagued by the same fear. Again, he takes one-third of the bananas in the pile and again the quantity is one greater than would allow an even split into thirds, so he hands the extra banana to the monkey and hides his share.

Still later, the final castaway gets up and repeats the exact same procedure, unaware that the other two have already done it. Yet again, he takes a third of the bananas and ends up with one extra, which he gives to the monkey. The monkey is most pleased.

When the castaways meet in the morning to divide the banana loot, they all see that the pile has shrunk considerably, but say nothing—they're each afraid of admitting their nighttime banana thievery. They divide the remaining bananas three ways, and end up with one extra for the monkey.

Given all this, how many bananas were there in the original pile? (Note: There are no fractional bananas in this problem. We are always dealing with whole bananas.)

**Answer**: 79. Note that if the pile were bigger, the next possible number that would meet the criteria above would be 160—but that's outside the scope listed in the second sentence ("between 50 and 100") of the puzzle.

*(Adapted from a brain teaser by Carl Proujan.)*

#### 21. THE VIRUS RIDDLE

In the video below, a virus has gotten loose in a lab. The lab is a single story building, built as a 4x4 grid of rooms, for a total of 16 rooms—15 of which are contaminated. (The entrance room is still safe.) There's an entrance at the northwest corner and an exit at the southeast corner. Only the entrance and exit rooms are connected to the outside. Each room is connected to its adjacent rooms by airlocks. Once you enter a contaminated room, you must pull a self-destruct switch, which destroys the room and the virus within it—as soon as you leave for the next room. You cannot re-enter a room after its switch has been activated.

If you enter via the entrance room and exit via the exit room, how can you be sure to decontaminate the entire lab? What route can you take? See the video for a great visual explanation of the problem and the solution.

**Answer**: The key lies in the entrance room, which is not contaminated and which you may therefore re-enter after exiting it. If you enter that room, move one room to the east (or the south) and decontaminate it, then re-enter the entrance room and destroy it on your way to the next room. From there, your path becomes clear—you actually have four options to complete the path, which are shown in the video above. (Sketching this one on paper is an easy way to see the routes.)

#### 22. THE IN-LAW CONUNDRUM

According to puzzle book author Carl Proujan, this one was a favorite of author Lewis Carroll.

The prime minister is planning a dinner party, but he wants it to be small. He doesn't like crowds. He plans to invite his father's brother-in-law, his brother's father-in-law, his father-in-law's brother, and his brother-in-law's father.

If the relationships in the prime minister's family happened to be arranged in the most optimal manner, what would be the *minimum possible number* of guests be at the party? Note that we should assume that cousin marriages are permitted.

**Answer**: One. It is possible, through some complex paths in the prime minister's family, to get the guest list down to one person. Here's what must be true: The PM's mother has two brothers. Let's call them brother 1 and brother 2. The PM also has a brother who married the daughter of brother 1, a cousin. The PM also has a sister who married the son of brother 1. The host himself is married to the daughter of brother 2. Because of all this, brother 1 is the PM's father's brother-in-law, the PM's brother's father-in-law, the PM's father-in-law's brother, and the PM's brother-in-law's father. Brother 1 is the sole guest at the party.

*(Adapted from a brain teaser by Carl Proujan.)*

#### 23. THE PRISONER BOXES RIDDLE

In the video, ten band members have had their musical instruments randomly placed in boxes marked with pictures of musical instruments. Those pictures may or may not match up with the contents.

Each member gets five shots at opening boxes, trying to find their own instrument. Then, they must close the boxes. They're not allowed to communicate about what they find. If the entire band fails to find their instruments, they'll all be fired. The odds of them randomly guessing their way through this is one in 1024. But the drummer has an idea that will radically increase their odds of success, to more than 35 percent. What's his idea?

**Answer**: The drummer told everyone to first open the box with the picture of their instrument. If their instrument is inside, they're done. If not, the band member observes what instrument is found, then opens the box with that instrument's picture on it—and so forth. Watch the video for more on why this works mathematically.

#### 24. S-N-O-W-I-N-G

One snowy morning, Jane awoke to find that her bedroom window was misty with condensation. She drew the word "SNOWING" on it with her finger. Then she crossed out the letter N, turning it into another English word: "SOWING." She continued this way, removing one letter at a time, until there was just one letter remaining, which is itself a word. What words did Jane make, and in what order?

**Answer**: Snowing, sowing, owing, wing, win, in, I.

*(Adapted from a brain teaser by Martin Gardner.)*

#### 25. THE MYSTERY STAMPS

While on vacation on the island of Bima, I visited the post office to send some packages home. The currency on Bima is called the pim, and the postmaster told me that he only had stamps of five different values, though these values are not printed on the stamps. Instead, the stamps have colors.

The stamps were black, red, green, violet, and yellow, in descending order of value. (Thus the black stamps had the highest denomination and yellow the lowest.)

One package required 100 pims worth of stamps, and the postmaster handed me nine stamps: five black stamps, one green stamp, and three violet stamps.

The other two packages required 50 pims worth each; for those, the postmaster handed me two different sets of nine stamps. One set comprised one black stamp and two each of the other colors. The other set was five green stamps, and one each of the other colors.

What would be the smallest number of stamps needed to mail a 50-pim package, and what colors would they be?

**Answer**: Two black stamps, one red stamp, one green stamp, and one yellow stamp. (It may help to write out the stamp formulas given above using the various b, r, g, v, and y. Because we know that b > r > g > v > y, and we have three described cases, we can do some algebra to arrive at values for each stamp. Black stamps are worth 18 pim, red are worth 9, green are worth 4, violet are worth 2, and yellow are worth 1.)

*(Adapted from a brain teaser by Victor Bryant and Ronald Postill.)*

Sources: *Brain Teasers* by Jan Weaver; *Brain Teasers & Mind Benders* by Charles Booth-Jones; *Riddles and More Riddles* by J. Michael Shannon; *Brain Teasers Galore: Puzzles, Quizzes, and Crosswords from Science World Magazine*, edited by Carl Proujan; *The Arrow Book of Brain Teasers* by Martin Gardner; *The Sunday Times Book of Brain Teasers*, edited by Victor Bryant and Ronald Postill.