Puzzle

Puzzle: A hotel with infinite rooms is fully occupied. A new guest arrives. Can the hotel accommodate the guest? Answer: Yes, by moving each current guest from room nnn to room n+1n+1n+1, the first room becomes available for the new guest. This demonstrates the counterintuitive properties of infinity.

Puzzle: You have three envelopes: one contains a check for $100, another $200, and another $300. You pick one at random and see that it has $200. You can keep it or switch. Should you switch? Answer: Switching doesn’t change your odds as each envelope was equally likely at the start. This puzzle illustrates how perceived probability doesn’t change once …

Puzzle: Prove that in any group of 367 people, at least two have the same birthday. Answer: By the pigeonhole principle, there are only 366 possible birthdays (including leap years). If there are 367 people, at least two must share a birthday, as the number of people exceeds the number of distinct days.

Puzzle: You have 12 coins, one of which is counterfeit, and a balance scale. The counterfeit coin is either heavier or lighter than the others. How can you identify the counterfeit coin and determine whether it is heavier or lighter in just three weighings? Answer: Divide the coins into three groups of four. Weigh two groups. If they balance, the …

Puzzle: In five houses, each with a different color, live five different people, each with a different nationality. They each have a different pet, drink a different drink, and smoke a different brand of cigarettes. Given various clues, who drinks water, and who owns the zebra? Answer: The solution is that the Norwegian drinks water, and the Japanese owns the …

Puzzle: Draw a house with five rooms. Is it possible to draw a path that crosses each wall exactly once? Answer: No, it is not possible. This is another Eulerian path problem where the degree of each vertex makes such a path impossible.

Puzzle: Can you cover an 8×8 chessboard with 31 dominoes, each covering two adjacent squares, if two opposite corners are removed? Answer: No, it is not possible. Removing two opposite corners results in two squares of the same color being removed. Each domino covers one black and one white square, but with an imbalance of colors (30 of one color, …

Puzzle: Three friends check into a hotel room that costs $30. They each pay $10 to the manager. Later, the manager realizes the room was only $25 and gives $5 to the bellboy to return. The bellboy, however, cheats and gives each friend $1 back, keeping $2 for himself. Now each friend has paid $9, totaling $27, plus the $2 …

100 prisoners are in solitary confinement, but there is one room with a light bulb that is initially off. The prisoners can visit the room one at a time in any order, and each time a prisoner enters, they can toggle the light bulb. If a prisoner concludes that all prisoners have visited the room, they can declare it. If …

A farmer needs to carry a wolf, a goat, and a cabbage across a river. He has a boat that can carry him and one other item at a time. If left alone, the wolf will eat the goat, and the goat will eat the cabbage. How can he carry all three across the river without anything being eaten? Answer: …

In a group of 23 people, what is the probability that at least two people share the same birthday? Answer: Surprisingly, the probability is about 50.7%. This is counterintuitive because we think of 365 days, but the number of possible pairs increases rapidly as people are added to the group, leading to a high likelihood of a shared birthday.

You have three rods and n disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top. The objective of the puzzle is to move the entire stack to another rod, obeying the following rules: Only one disk …

The city of Königsberg (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands connected to each other and the mainland by seven bridges. The problem is to find a walk through the city that would cross each bridge once and only once. Answer: It is not possible to cross each bridge exactly …

You’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say Door 1, and the host, who knows what’s behind the doors, opens another door, say Door 3, which has a goat. He then asks you if you want to switch to Door …

You find yourself in a mysterious room with three doors. Each door leads to a different path, but only one path will take you to safety. The other two paths are extremely dangerous and will lead to certain doom. In front of each door, there is a guard. You know the following about the guards: One guard always tells the …