1) Strange Trip

You travel 100 miles north, 100 miles east, and then 100 miles south. You are at the same point that you started from. Describe all the places on earth this could be, if any.

2) Three Cards

There are 3 cards: one is all red, one is all blue, and the third is blue on one side and red on the other. The cards are shuffled. You pick one at random and it is blue on the side facing you. What are the chances that it is also blue on the other side?

3) Monty Hall Problem

Behind 1 of 3 closed doors is a prize. You pick one of the doors. Monty opens one of the other doors and it is empty. You are given the choice of sticking with your choice or switching to the other unopened door. Should you switch? What are your chances of winning if you do?

4) Door Monty Hall Problem

Behind 1 of 4 closed doors is a prize. You pick door 1. Monty opens the 4th door and it is empty. You switch your choice to door 2. Now monty opens door 3 and it is also empty. Given the choice, should you switch back to door 1, and what are your chances of winning if you do?

5) Two Envelopes

(Also known as the exchange paradox.) There are two envelopes, one contains twice as much money as the other. You pick one at random and find $100 inside. What is the "expected value" of the amount in the other envelope? Given the option, should you switch? Does it matter how much you found in that first envelope? Does it matter if you open the envelope?

What's Next?  

What is the next figure in this sequence?

6) Helium

A helium balloon is in your car. How does it move when you slam on the brakes?

7) Urns with Balls

There are 2 urns filled with 100 ping pong balls total. 50 ping pong balls are white and 50 are black. I reach randomly into one of the urns, stir, and pick one ball without looking. What are the highest possible chances of me picking a white one that you can cause if you arrange the balls in the urns ahead of time?

8) Bean Bag

A bag contains a single bean, known to be either white or black. A new white bean is added to the bag, and it is shaken. A bean is taken back out, and it is white. What are the chances the remaining bean is also white?

9) Four Trees

A farmer claims to have 4 trees on his land all equidistant from each other. Could this be true?

10) Suicidal Spots

In a far away land, there is an unusual tribe of 300 perfectly logical and perfectly intelligent people. Each member has a visible spot on the back of his or her head, some are red and some are black. Nobody knows the color of their own spot, but they do know the color of everyone else's. If a tribesman ever realizes the color of his own spot it is strict custom that he publicly commit suicide the following morning, so they never mention spot colors, and have no mirrors. But then one day an American tourist visits this land and announces to the entire tribe: "I can see that at least one of you has a red spot." The tourist leaves and returns a year later. What has happened?

11) One Question

You are shipwrecked on an island. There is a fork in the path to the other side, one way leads to a safe village, the other leads to hungry cannibals. There are twin brothers who both know which path is which, but one of the brothers is honest, and the other always lies. You may ask one of them a single question. What should it be? What question would you ask if there is just one person who is honest or lies but you don't know which?

12) Random Hats

Three people are given hats. Each hat is either red or blue, chosen at random. Each person can see the other 2 hats, but not their own. They each must simultaneously either guess their own hat's color, or pass. No communication is allowed, although they can agree on a strategy ahead of time. What strategy will give them the best chances of at least one person guessing right, and nobody guessing wrong?

13) Northwest Spiral

Two pilots fly from the Equator to the North Pole. The first flies north in a straight path. The second flies on a spiral path by always heading northwest. How far does the second pilot travel? (relative to the first)

14) Odd Ball

There are 12 equal sized balls. One ball has a slightly different weight (more or less) than the other 11. Can you use a balance scale only 3 times to find the odd ball?

15) Bags of Gold

Ten bags each contain nine pieces of gold. The gold pieces are all supposed to weigh 1 ounce, but the pieces in one bag weigh only .9 ounces. Use an accurate scale just once to find which bag contains the lighter pieces.

16) Fair Cake

When two people want to share a cake fairly, one cuts, and the other chooses. Assuming this is a fair scheme, devise a similar scheme for 3 people and 1 cake. Nobody should get short caked even if the other 2 cooperate.

17) Light Switches

There are 3 incandescent light bulbs in one room and 3 switches for these bulbs in another room. No light from the bulbs can be seen outside of their room. You are allowed to enter the room with the bulbs only once. How can you figure out which switches are connected to which bulbs?

18) Grid of Tiles

There is an empty 8x8 grid, except two opposite corners are missing. Can you tile the rest with 1x2 tiles?

19) 1x3 Tiles

Can you cover an 8x8 grid with 1x3 tiles and a single 1x1 tile? In what locations can the 1x1 tile be?

20) Cheese Cubes

A block of cheese is cut into a 3x3x3 grid of subcubes. A mouse starts eating one corner, and moves on to adjacent pieces until they are all eaten. Could he eat the middle piece last?

21) Measuring with Jugs

Using a 5 liter jug, a 3 liter jug, and a hose, can you measure 4 liter of water? How about measuring 1 liter?

22) Burning Fuses

You have 2 lengths of fuse and 2 matches. One fuse will burn from start to end in 10 minutes, and the other in 15 minutes. However, their burn rate is not steady along their lengths. Measure 20 minutes of time.

23) Pocket Change

If all the coins in my pocket except two are pennies, all except two are nickels, and all except two are dimes, how much money do I have?

24) Gloves and Germs

You have 3 cultures of highly contagious and deadly germs. As part of an important experiment you must squeeze each culture once with your entire hand. However, you have only 1 pair of latex gloves. You must not contaminate any culture with another or with germs from your skin. The gloves can be worn on either the left or right hand. How can you complete your experiment without contaminating anything?

25) Mixed Up Liquids

There are two equally filled jars, one contains milk, the other water. A teaspoon of milk goes into the water and is stirred. Then a teaspoon of the mixture goes back into the milk. Is there more water in the milk or milk in the water?

Feynman's Sucking Sprinkler

          If you take a water sprinkler like the one above and put it under water, it will spin clockwise as it would on land. But what happens if you then reverse the flow so it sucks in water? Would it spin, and if so, in which direction?

26) Train Full of Water

A train car is at rest on a frictionless track. The car is full of water and has a spout pointing downwards on the far right end. The spout is opened and the water pours out. Describe the movement of the car, if any.

27) Bubbles in Space

Two balls of matter in empty space would move towards each other. What if all space were filled with a frictionless deformable substance, except for two empty bubbles? How would the bubbles move?

28) Six Chop Sticks

Using 6 equal length chop sticks make exactly 4 equilateral triangles.

29) Stick Boxes

          Make 4 equal sized squares out of these 5 by moving only 2 sticks, and leave no extra sticks.

30) Fish Sticks

                    Point the fish the other way by moving only 3 sticks.

31) Ten Points

Can you arrange 10 points so that 5 lines can each be drawn through 4 points?

32) Connect the Dots

                    Connect these 9 dots with only 4 connected straight line segments. (Don't lift your pencil.)

33) Wire Cube

What is the minimum number of wires needed to make a cube? (Bend but don't double the wires.)

34) Bookworm

A encyclopedia with ten 200 page volumes is sitting on a bookshelf in the usual order. A bookworm starts on the first page and eats in a straight line to the last page. How many total pages does he eat through?

35) Five Hats

Three wise men are lined up in single file, each wearing a white hat. They know there were 5 hats total, three white, and two black, but they can only see the hats on the men in front of them. They don't speak unless they figure out what color hat they have on. Who figures out first?

36) Pop Quiz

A teacher announces there will be a surprise quiz sometime during the week. The students argue it can't be on Friday because if the teacher waits until the last day it won't be a surprise anymore. Once they know it can't be Friday they argue by the same reasoning that it can't be Thursday either. What is wrong with this logic?

37) Darts

You throw two darts at a dart board, aiming for the center. The second lands farther from the center than the first. You then throw another dart at the board, aiming for the center. Assume your skill level is consistent. What are the chances that this third dart also lands farther from the center than the first?

38) Girl Babies

A large tribe obeys a strict reproductive custom. All families continue having children until they have a girl, and then they stop having more children. Assume it is equally likely for a given birth to produce a girl or boy, and assume families can have any number of children so they always do get one girl eventually. In the 10th generation, what is the expected ratio of males to females? Also, what is the expected population size of the 10th generation relative to the 1st?

39) Hotel Bellboy

Three people check into a hotel. They pay $30 to the manager and go to their room. The manager finds out that the room rate is $25 and gives $5 to the bellboy to return. On the way to the room the bellboy reasons that $5 would be difficult to share among three people so he pockets $2 and gives $1 to each person. Now each person paid $10 and got back $1. So they paid $9 each, totaling $27. The bellboy has $2, totaling $29. Where is the remaining dollar?

40) Doctor Who

A father and his son are in a car crash. The father is killed instantly but the son is only injured and is taken to the hospital. He is rushed to the operating room, the doctor comes in, looks at the patient on the operating table, and says, "I can't operate on him, he's my son." How can this be?

41) Non-Self Containing Sets

What is difficult about the set of all sets that do not contain themselves?

42) Manhole Covers

Why are manhole covers round?

43) Mirrors

Why do mirrors seem to flip things left-right but not up-down?

44) Centrifuge

Can you spin 5 samples in a 12-hole centrifuge, without it being out of balance?

45) 1000 Doors

A thousand doors start closed. Someone walks along and changes the state of each door (opens or closes). A 2nd person changes the state of every 2nd door, and a 3rd person changes the state of every 3rd door. This continues until the 1000th person changes the state of the 1000th door. How can you know if the Nth door is now open or closed? (without simulating the whole thing)

46) Square of Bugs

Four bugs are on the corners of a 1 meter square. Each bug always faces the next bug (on the next clockwise corner). If they all walk forward at the same speed until they meet, how far does each bug travel?

47) A Bee and Two Trains

Two trains are 30 miles apart, and travel towards each other at 5 mph and 10 mph. A bee starts at the slower train and flies at 25 mph to the other train. Each time it reaches a train it turns around and flies back to the other train again. What is the sum of the distances that the bee has flown when the trains meet?

48) Backwards Bee and Two Trains

Two trains start end-to-end at the same point and travel away from each other at 5 mph and 10 mph. A bee also starts at the same point and flies back and forth at 25 mph between the ends of the moving trains. After 2 hours, where is the bee?

49) Water Levels

You are on a boat in a small pond. You have a stone and a log in the boat. You throw the stone into the water. Does the water level in the pond rise, fall or stay the same? How about if you throw the log in?

50) Pieces of Stone

A farmer has a 40 lb stone that he uses to measure out bales of hay on a 2 sided balance. He loans it to a friend who accidentally breaks it into 4 pieces. Instead of being angry, the farmer is quite happy. He says to the friend, "you managed to break it into just the right 4 pieces that will now let me weigh any weight between 1 and 40." What are the weights of the 4 pieces?

51) Breaking Balls

You have 2 bowling balls that each breaks when dropped from the same height. You want to find the highest floor of a 100 story building from which these balls can be dropped without breaking. Devise an optimal procedure that can always locate that floor using not more than N drop tests. What is the smallest N can be?