Tricky Yet Fun Math Questions With Solutions

 These problems challenge your problem-solving skills, logical reasoning, and ability to think outside the box.

The Missing Dollar Puzzle

Problem: Three friends check into a hotel room that costs $30. They each contribute $10. Later, the hotel manager realizes the room was only supposed to cost $25, so she gives $5 to the bellboy to return to the friends. The bellboy, however, decides to keep $2 as a tip and gives $1 back to each friend. Each friend has paid $9, and 3 × 9 = $27. The bellboy has $2, so $27 + $2 = $29. What happened to the missing dollar?

Solution: The confusion arises from the way the problem is framed. The friends paid $27 in total. Of that $27, $25 went to the hotel for the room, and the remaining $2 was kept by the bellboy as a tip. There is no missing dollar; it's simply a misdirection caused by adding the tip to the total amount paid.

The Bridges of Konigsberg

Problem: Seven bridges connect different parts of the city of Konigsberg. The problem was to find a walk that would cross each bridge exactly once. Is it possible?

Solution: This famous problem was solved by mathematician Leonhard Euler in 1736. Euler showed that it is impossible to walk through the city crossing each bridge only once. His solution was based on graph theory, where the parts of the city are vertices and the bridges are edges. Euler concluded that to have such a walk, all but two vertices must have an even degree (an even number of edges). Since in Konigsberg, there were four vertices with an odd degree, it is impossible to walk across each bridge only once.

The Monty Hall Problem

Problem: You’re on a game show, and the host Monty Hall offers you three doors: behind one is a car, behind the other two are goats. You choose a door, and Monty, who knows what’s behind each door, opens another door revealing a goat. He then allows you to switch your choice to the remaining door. Should you switch, stick with your original choice, or does it not matter?

Solution: The best strategy is to constantly switch. Initially, the probability of picking the car is 1/3, and the likelihood of choosing a goat is 2/3. After Monty opens a door with a goat, the probability of the car being behind the remaining door increases to 2/3, while the probability of the car being behind the door you originally picked remains 1/3. Therefore, switching increases your chances of winning.

1. The Two Envelopes Paradox

Problem: You are given two envelopes, each containing an unknown amount of money. One envelope contains twice the amount of money as the other. You are asked to pick one envelope and are told that if you switch, you can expect to win more money. Should you switch?

Solution: The paradox arises because of a faulty argument that switching always seems better. Let the amount in the first envelope be xx. The second envelope then contains either 2x2x or x/2x/2, each with a probability of 50%. The expected value for switching is:

E[switch]=12(2x)+12(x2)=x+x4=5x4.E[\text{switch}] = \frac{1}{2}(2x) + \frac{1}{2}\left(\frac{x}{2}\right) = x + \frac{x}{4} = \frac{5x}{4}.This suggests that switching is always beneficial, but this conclusion is flawed due to the nature of expected value and the lack of information about the actual amounts. The paradox highlights the importance of considering how probability is used in decision-making.

The Four 4’s Problem

Problem: Using precisely four 4's and any mathematical operations, express the numbers 1 to 10.

Solution: Here’s how we can represent each number:

• 1 = (4 / 4)
• 2 = (4 / 4) + (4 / 4)
• 3 = (4 + 4) / 4
• 4 = 4
• 5 = (4 × 4) / 4
• 6 = (4 + 4) / (4 / 4)
• 7 = (4 + 4) - (4 / 4)
• 8 = 4 + (4 + 4) / 4
• 9 = (4 + 4) + (4 / 4)
• 10 = (4 × (4 + 4)) / 4

This problem is an interesting exercise in mathematical creativity.

These puzzles not only provide fun but also offer valuable insights into problem-solving and mathematical thinking. They challenge you to think logically, explore different angles, and appreciate the elegance of mathematics.