Gurmeet.Net
Puzzles

## Bigger or Smaller

### Puzzle

Alice writes two distinct real numbers between 0 and 1 on two chits of paper. Bob selects one of the chits randomly to inspect it. He then has to declare whether the number he sees is the bigger or smaller of the two. Is there any way he can expect to be correct more than half the times Alice plays this game with him?

### Source

Heard from Gagan Aggarwal in 2000--2002.

### Solution

Let the number revealed to Bob be p. Then Bob should say "bigger" with probability p, "smaller" otherwise. If the other number is q, then the probability of winning is ½ + ½ |p - q|.

Previous Puzzle: Forks in a Road

A traveler has to pass tests of increasing difficulty to meet an Eastern mystical master. For the first test, he meets a pair of twins at a fork in the road: one path leads to the jungle, the other to the mystic. One of the twins always says the truth, the other always lies. What yes/no question should you ask one of the twins to determine the path that goes to the mystic? For the second test, the traveler encounters a second fork in the road. Again, one path leads to the jungle, the other to the mystic. This time, there are three look-alike brothers: one always tells the truth, the second always lies but the third sometimes tells the truth and sometimes lies. What two Yes/No questions should the traveler ask two of the brothers to determine the path to the mystic? Each question is answered by only the brother it is posed to. For the second question, the traveler may choose the brother and the question depending upon the answer to the first question.

Next Puzzle: Measuring Weights

(a) Customers at a grocer's shop always want an integral number pounds of wheat, between 1 pound and 40 pounds. The grocer prefers to measure wheat in exactly one weighing with a beam balance. What is the least number of weights he needs? (b) Customers come to a pawn shop with antiques. An antique always weighs an integral number of pounds, somewhere between 1 pound and 80 pounds. The owner of the pawn shop is free to do as many weighings as necessary to ascertain the unknown integral weight by using a beam balance. What is the least number of weights he needs?