Dwight Lahr and David Rudel

Friday Discussion: Week #2

1. We discussed the notion of modular arithmetic, and how, in modulo 9 for example, there was a number that acted like “1/2” in the sense that you could multiply it by 2 to get 1. What other things make sense in mods, and what don’t? For example:

i. Does “square root of 2” make sense in mod 7?

ii. What about “the logarithm to the base 3 of 4” in mod 11?

iii. Does 3! (factorial) make sense in modulo 10?

2. There is a convention of sheep, and each sheep at the convention wears either a red hat or a blue hat. At the beginning of the convention, the keynote speaker says, “I see some of you are wearing blue hats and some are wearing red hats.” [The speaker is and can be trusted.] Unfortunately, it is a very bad thing in sheep culture to wear a blue hat, and in fact any sheep that realizes he is wearing a blue hat will immediately run away at night. Sheep may not look in a mirror, ask each other what hat they are wearing, etc. However, they are perfectly logical. After 14 days 10 sheep have left the convention. How many have left the convention after 21 days?

The next ten problems are taken from a book called “What is the Name of This Book?” by Raymond Smullyan. The problems concern the island of knights and knaves. On this island, some of the inhabitants, called knights, always tell the truth, while others, called knaves, always lie. You are a tourist on the island and you stop to talk to some of the inhabitants.

1. You meet three inhabitants of the island, A, B, and C. You ask A, “Are you a knight or a knave?” A answers, but he mumbles and you can’t hear what he says. You ask B, “What did A say?” B replies, “A said that he is a knave.” C then says, “Don’t believe B, he is lying!” What are B and C?

2. You travel a little farther and meet another three inhabitants, A, B, and C. This time you ask A, “How many knights are among you?” but again you can’t make out his reply. So you ask B, “What did A say?” and B replies, “A said there is one knight among us.” Again C says, “Don’t believe B, he is lying!” Again, what are B and C?

3. A little farther on in your travels you meet two people, A and B. A tells you, “At least one of us is a knave.” What are A and B?

4. Suppose A had said, “Either I am a knave or B is a knight.” What are A and B?

5. Again there are three people, A, B, and C, each of whom is either a knight or a knave. You hear them remark:

A: All of us are knaves.

B: Exactly one of us is a knight.

What are A, B, and C?

6. Suppose instead A and B say the following:

A: All of us are knaves.

B: Exactly one of us is a knave.

What can you say about A, B, and C?

7. Suppose A says, “I am a knave, and B is not a knave.” What are A and B?

8. Again there are three inhabitants. A says “B is a knave,” and B says, “A and C are of the same type.” (i.e. both knights or both knaves) What is C?

9. Again three people A, B, and C. A says, “B and C are of the same type.” You then ask C, “Are A and B of the same type?” What does C answer?

10. You come across two inhabitants of the island, A and B, resting under a tree. You ask A, “Is either of you a knight?” He answers, and you know the answer to your question. What are A and B?