Fish in a Pond

There are 5 fish in a pond. What is the probability that you can split the pond into 2 halves using a diameter, so that all fish end up in one half?

Let us generalize the problem to N fish in a pond. We can assume that all fish are on the boundary of the pond, which is a circle, and we need to find the probability that all of them are contained within a semi-circle.

For every fish Fᵢ, consider the semi-circle Cᵢ whose left end-point is at Fᵢ. The probability that all fish belong to Cᵢ is equal to 1/2ᴺ⁻¹. Since it is impossible to have 2 fish Fᵢ and Fⱼ, such that the semi-sircles Cᵢ and Cⱼ contain all fish, we see that the probability that all fish belong to Cᵢ for some i is equal to N/2ᴺ⁻¹.

When N = 5, we get that the answer is 5/16.

Pronunciation Puzzles

The following 2 puzzles rely on misleading phrasing of the questions. Read them aloud to your friends and let them ponder upon them.

  1. What has 4 letters, sometimes 9, and never 5
  2. One night a king and a queen stayed in a hotel. On the next day, three people checked out. Who was the third person?
  3. There are 30 cows and 28 chickens. How many didn’t?
  4. Pronounce the following words: T-W-A, T-W-E, T-W-I, T-W-O
  5. As I was walking across the London Bridge, I met a man.
    He tipped his hat, and drew his cane.
    In this riddle, I said his name. What is it?

The first puzzle is not a question. It is a statement, saying that the word “what” has 4 letters, the word “sometimes” has 9 letters, and the word “never” has 5 letters. There is nothing to solve, so the puzzle is figuring that out!

The second puzzle reads as “One knight, a king, and a queen stayed in a hotel.” Thus, the third person was the knight.

The third puzzle reads as “There are 30 cows and 20 ate chickens. How many didn’t?” Thus, the answer is that 10 cows didn’t eat chickens.

The fourth question often confuses people and they pronounce TWO as [twou] instead of [tuː].

The fifth riddle actually says: “He tipped his hat, ‘Andrew Hiscane'”. Thus, the name of the man is Andrew Hiscane.

Rapunzel and the Prince

The evil witch has left Rapunzel and the prince in the center of a completely dark, large, square prison room. The room is guarded by four silent monsters in each of its corners.  Rapunzel and the prince need to reach the only escape door located in the center of one of the walls, without getting near the foul beasts. How can they do this, considering they can not see anything and do not know in which direction to go?

The prince must stay in the center of the room and hold Rapunzel’s hair, gradually releasing it. Then, Rapunzel must walk in circles around the prince, until she gets to the walls and finds the escape door.

Is This Prime?

In the past few days, I, my friends, and a lot of Twitter people have been trying to beat each other’s scores in the game “Is This Prime?”.

The game itself is simple; you are shown random integers on the screen, and you need to guess whether they are prime or composite. It is fast-paced and fun, and a good opportunity to exercise some mental math. Below we have listed a few tips which can help you beat our personal record of 67 points.

  1. Memorize as many numbers as possible. Knowing the multiplication table up to 10×10, it should be easy to learn by heart whether each number up to 100 is prime or composite.
  2. Pay attention to the last digit. If it is 5, then the number is composite (unless it is =5).
  3. Check whether the sum of the digits is divisible by 3. If it is, then the number is composite (unless it is equal to 3).
  4. If the number is between 100 and 300, check whether the sum of the first and the third digits equals the second digit. If this is true, then the number is divisible by 11, and therefore it is composite. 209 is the only other number in this range divisible by 11.
  5. Remember the sneaky composites: 119, 133, 161, 169, 247, 253, 259, 289, 299.

The Puzzle TOAD

The Puzzle TOAD is a website, created by four Carnegie Melon professors (Tom Bohman, PO Shen-Loh, Alan Frieze, Danny Sleator), where you can find a growing collection of ingenious math brain teasers. Unlike Puzzle Prime, The Puzzle Toad is targeted exclusively towards math and computer science majors. Students who are preparing for college Olympiads will find the problems particularly useful. Check out The Puzzle TOAD by clicking the banner below.