Puzzle: At a party with 20 people, if each person shakes hands once with every other person, how many handshakes occur?

Answer:

Puzzle:
Prove that it is impossible to walk through the city of Königsberg and cross each of its seven bridges exactly once.

Answer:
Euler’s solution involves graph theory: each landmass is a node, and bridges are edges. A walk is possible only if zero or two nodes have an odd number of edges. Königsberg has four nodes with odd edges, making it impossible.

Puzzle: Arrange the numbers 1-9 in a 3×3 grid such that each row, column, and diagonal sums to 15.

Answer: One solution is:

Puzzle: Two trains are 100 miles apart and move towards each other, each at 50 miles per hour. A fly starts from one train, flying at 75 miles per hour to the other, and turns around each time it reaches a train. How far does the fly travel before the trains collide?

Answer: The fly travels 75 miles. The trains meet in 1 hour, and in that time, the fly travels continuously at 75 miles per hour.

Puzzle: A camel must transport 3,000 bananas 1,000 km to a market. It can carry at most 1,000 bananas at a time and eats one banana per km. How many bananas can the camel deliver?

Answer: The camel can deliver a maximum of 533 bananas. By calculating the optimal strategy of carrying loads, dropping them, and returning for more, it minimizes banana consumption on trips.

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 $n$ to room $n+1$, 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 the choice is made, similar to Monty Hall without the host’s influence.

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: 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 zebra. This is solved using logical deduction based on the clues provided.

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, 32 of another), full coverage is impossible.

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 the bellboy kept makes $29. Where is the missing dollar?

Answer: There is no missing dollar. The total is still $30. The friends paid $27, of which $25 went to the hotel and $2 to the bellboy. The $9 calculation includes the bellboy’s $2, which is not needed. The $25 payment to the hotel and $3 returned to the friends account for the full $30.

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 correct, they are freed; if wrong, they are executed. How can they be freed?

Answer: Designate one prisoner as the counter. Only this prisoner can turn the light bulb off. Every time a different prisoner enters and the bulb is off, they turn it on if they haven’t done so before. The counter counts each time they turn the bulb off. Once they count 99 times, they can declare everyone has visited.

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:

1. Take the goat across the river first.
2. Return alone and take the cabbage across.
3. Leave the cabbage and bring the goat back.
4. Take the wolf across the river.
5. Return alone and finally take the goat across.