Answers to last issue's Brain Teasers

1. Geetha was baking biscuits and suddenly found her ring missing. She had baked 9 biscuits in that batch so she knew that the ring must have fallen in to one of them. She doesn't want to crumble up all the biscuits but she certainly does not want to lose her ring. She has a weighing machine but no weights. How can she find the ring? What is the smallest number of times she needs to use the weighing machine in order to find the ring-biscuit?

Answer: She can weigh the biscuits against each other. The most efficient way is to weigh any three biscuits against any other three. If their weights are equal, then one of the remaining two has the ring. Weigh these against each other: the heavier one has the ring. 
	If the three pairs of biscuits have different weights, then one set is heavier than the other. One of these three is the one containing the ring. Weigh any two of them against each other. If they are the same, the third biscuit has the ring. If they are difrerent, the heavier one has the ring.
	So the minimum number of times you need to weigh is twice.

2. Knights and Knaves is a type of logic puzzle devised by the famous logician Raymond Smullyan. On a fictional island, all inhabitants are either knights or knaves. Knights always tell the truth, while knaves always lie.
	The puzzles involve a visitor to the island who meets John and Bill, both residents of the island.
	John says: We are both knaves.
	Question : Is John a knight or a knave? Is Bill a knight or a knave?
	Here is perhaps the most famous of this type of puzzle: John and Bill are standing at a fork in the road. You know that one of them is a knight and the other a knave, but you don't know which. You also know that one road leads to Death, and the other leads to Freedom.
	Question : By asking one yes/no question, can you determine the road to Freedom? By asking one yes/no question, can you determine whether John is a knight?

	Answer: This is what John is saying in a more extended form:
"John is a knave and Bill is a knave."
	If John was a knight, he would not be able to say that he was a knave since he would be lying. Therefore the statement "John is a knave" must be true.
	Since knaves lie, and one statement is true, the other statement must be false. Therefore the statement "Bill is a knave" must be false which leads to the conclusion that Bill is a knight.
	The solution is that John is a knave and Bill is a knight.
	For finding out which way leads to freedom the following question should be asked; "Will the other man tell me your path leads to Freedom?"
	If the man says Yes then the path doesn't lead to Freedom, if he says No then it does. The following logic is used to solve the problem. In this instance John is the Knight. You go up to him and ask him the question. If his path does lead to freedom he will say No since the Knave would lie and say the Knight's path doesn't lead to freedom. If his path doesn't lead to freedom he will say Yes since the Knave would say the path leads to freedom.
	In this instance John is the Knave. You go up and ask him the question. If his path does lead to freedom he will say No since the Knight would say it does lead to freedom. If his path doesn't lead to freedom he would say Yes since the Knight would tell you it doesn't lead to freedom.
	The simplest solution to finding out which one is the Knight is to ask any obvious question, such as: Is 2+2 = 4? Is a foot 12 inches? etc. The Knight would reply Yes, and the Knave would reply No.