Puzzles
#001 Monty Hall
You are on a game show. The host presents you with three doors. Behind one door is a car; behind the other two are goats. You pick a door. The host, who knows what's behind each door, opens a different door to reveal a goat. He then asks you: do you want to switch to the other unopened door, or stick with your original choice?
Should you switch?
#002 Blue Eyes
On an island, there are 100 people with blue eyes, 100 with brown eyes, and one guru with green eyes. They can all see everyone else's eye color, but they do not know their own. There are no mirrors, and no one ever discusses eye color. Every night, a ferry comes to the island. If anyone has figured out their own eye color, they must leave on the ferry that night.
One day, the guru makes a single public announcement, heard by everyone on the island: "I can see someone with blue eyes."
Who leaves the island, and on what night?
#003 Twelve Spheres
You have 12 identical-looking spheres. Exactly one of them is a different weight from the rest - but you don't know whether it's heavier or lighter. You also have a balance scale, which you can use exactly 3 times.
Find the odd sphere, and determine whether it is heavier or lighter.
#004 Twenty-Five Horses
You have 25 horses and a racetrack that fits 5 horses at a time. Racing them only tells you the relative finishing order of those 5 horses.
What is the minimum number of races needed to find the 3 fastest horses? You do not have a clock.
#005 Lions and Sheep
There are 100 lions and 1 sheep on an island. A lion would like to eat the sheep, but if a lion eats a sheep, it becomes a sheep itself, then becoming vulnerable to being eaten by another lion. All lions are perfect logicians and act in their own self-interest. A lion would rather not eat than risk being eaten.
For which values of N does the sheep get eaten?
#006 Two Guards
You arrive at a fork in the road. One path leads to freedom, the other to certain death. Two guards stand at the fork. One always tells the truth; the other always lies. You don't know which is which.
You may ask one guard a single yes-or-no question. What do you ask to determine the correct path?
#007 Chessboard Dominoes
Take a standard 8x8 chessboard and remove two diagonally opposite corners. You have 31 dominoes, each of which covers exactly two adjacent squares.
Can you tile the remaining 62 squares with the 31 dominoes?
#008 100 Prisoners and Boxes
A room contains 100 closed boxes, numbered 1 to 100. Inside each box is a slip of paper with a number from 1 to 100, each appearing exactly once. 100 prisoners, also numbered 1 to 100, must each enter the room one at a time and open at most 50 boxes. If every prisoner finds the slip with their own number, they all go free. Otherwise, they all die.
The prisoners may agree on a strategy beforehand, but once the process begins, they cannot communicate. Each prisoner leaves the room exactly as they found it.
Is there a strategy that gives them a reasonable chance of survival?
#009 Pirates and Gold
Five perfectly logical pirates (ranked by seniority: A, B, C, D, E) have found 100 gold coins. They must divide the loot according to the following rules:
- The most senior pirate proposes a distribution. - All pirates vote (including the proposer). If at least half vote in favor, the distribution is accepted. - If the proposal is rejected, the proposer is thrown overboard and the next most senior pirate makes a proposal.
Each pirate's priorities, in order, are: (1) survive, (2) maximize gold, (3) throw others overboard for fun, all else being equal.
What distribution does pirate A propose?
#010 100 Hats
100 prisoners stand in a single-file line, each wearing either a red or blue hat. Each prisoner can see all the hats in front of them but not their own or any behind them. Starting from the back of the line, each prisoner must guess the color of their own hat. They can only say "red" or "blue", and everyone in the line hears the guess.
The prisoners may agree on a strategy beforehand. What is the maximum number of prisoners that can be guaranteed to guess correctly?
#011 Burning Ropes
You have two ropes. Each rope takes exactly one hour to burn from end to end, but they burn at non-uniform rates - different sections of a rope may burn faster or slower than others. You have a lighter.
How do you measure exactly 45 minutes?
#012 Poisoned Wine
A king has 1000 bottles of wine. Exactly one bottle has been poisoned. The poison is undetectable and takes exactly 24 hours to kill. The king has 10 prisoners he's willing to use as testers. He needs to identify the poisoned bottle in exactly one round of testing (24 hours).
How can the king determine which bottle is poisoned?
#013 Prisoners and a Light Switch
100 prisoners are told that they will each be brought to a room, one at a time, in a random order (with repetition - the same prisoner may be brought back multiple times). The room contains a single light switch, initially off. At any point, a prisoner may declare: "All 100 of us have visited this room." If the declaration is correct, all prisoners go free. If it's wrong, they all die.
The prisoners may agree on a strategy beforehand but cannot communicate once the process starts. The only information available is the state of the light switch.
What strategy guarantees their freedom?
#014 Ants on a Pole
100 ants are placed at random positions on a 1-meter pole, each facing a random direction. They all start walking at the same speed (1 cm/s). When two ants collide, they both immediately reverse direction. An ant that reaches the end of the pole falls off.
How long until all ants have fallen off the pole?