Let L be the number of lakes.

• Total rivers flowing out: 3L (since each lake has 3 out).

• Total rivers flowing in: 4L (since each lake has 4 in).

But every river flows out of exactly one lake and into exactly one lake. Therefore, the total number

of rivers flowing out must equal the total number of rivers flowing in.

3L = 4L ⇒ L = 0

This is a contradiction. Fibland would need to have 0 lakes, which is nonsense.

Dmitri was wrong.

Consider the orientation of the triangle formed by the three pucks. Label the pucks so their initial order (say, clockwise) is A-B-C. The operation of hitting one puck through the other two is equivalent to reflecting that puck across the line joining the other two. Reflection reverses orientation.

• Each hit (reflection) flips the orientation (clockwise ↔ counterclockwise).

• The composition of an even number of reflections preserves orientation.

• The composition of an odd number of reflections reverses orientation.

The player performs this operation 25 times. Since 25 is odd, the final orientation of the pucks will be the opposite of the initial orientation (e.g., A-C-B instead of A-B-C). For the pucks to return to their exact starting positions, including their cyclic order, the orientation must be the same. Since it is reversed, they cannot be in the exact same configuration. No — The final configuration will have reversed orientation.

1. I met a resident of Mirkwood who said "I am an elf or I am sane". Exactly what type was he?

A: The statement is either true or false. If it is false, then he is neither an elf nor sane. So he must be an insane man. But then the statement must be true. This is a contradiction so the statement is true. In that case, he is either an elf or he is sane. If he is not an elf, then he must be a sane man. But sane men do not tell the truth, so he must be an elf. In that case, he must be a sane elf.

2. Another resident said, "I am not a sane elf". What type was he?

A: The statement is either true or false. If it is false, then he is either an insane elf or a sane man. In either case, the statement is true. This is a contradiction. Therefore, the statement is true. Hence, he must be an insane man.

3. Another resident said, "I am an insane elf". Is he the same type as the previous resident?

A: Again, the statement is either true or false. If the statement is true, then he would be an insane elf. But then the statement would be false. This is a contradiction. So the statement must be false. If it is false, then he is one of (sane elf/sane man/insane man). If he is a sane elf or an insane man, then the statement could not be false, so he must be a sane man. 

4. I once asked a resident, "Are you an insane man?". He answered "Yes" or "No" and I knew what he was. What was he?

A: A sane elf would have answered "No" to the question and all other types would have said "Yes". If he had said "Yes", then I would not know what he was. Since I knew what he was, he must have answered "No". So he was a sane elf.

5. I once met a resident of Mirkwood who said "I am a man." Can it be inferred whether he is an elf or a man? Can it be inferred whether he is sane or not?

A: A sane elf would not say that he is a vampire, and a sane man would lie about him being a man. Therefore, the resident is insane. However, we cannot determine whether he is an insane elf or an insane man.

1. I asked a resident of Mirkwood if Sauron was alive or dead. He said, "If I am an elf, then Sauron is still alive." Could it be determined if Sauron is alive?

A: No, it cannot. If he is an insane vampire, then the statement is true. Since he is not an elf, it is vacuously true, so the statement "Sauron is still alive" could be either true or false.

2. Suppose a resident said, "If I am either a sane elf or an insane man, then Sauron is still alive." Could it be determined if Sauron is alive?

A: Yes, it can be determined if Sauron is still alive. If he is sane elf, then the statement is true so Sauron is alive. If he is an insane elf, then the statement is false. The only way this is possible is if Sauron is dead. If he is a sane man, once again the statement is false so Sauron is dead. If he is an insane man, then the statement is true, so Sauron is alive.

3. Can I ask a single question and find out from a resident of Mirkwood whether or not he is an elf?

A: Ask him "Are you sane?". If the answer is "Yes", then he is an elf. If he answers "No", then he is a man.

4. Can I ask a single question and find out from a resident of Mirkwood whether or not he is sane?

A: Ask him "Are you an elf?". A sane resident will say "Yes" (regardless of whether he is an elf or a man), and an insane resident will say "No".

5. Can I ask a single question and find out from a resident of Mirkwood if Sauron is still alive?

A: There are many such questions. For instance, you could ask "Is the statement `You are a sane elf or an insane man' equivalent to the statement 'Sauron is alive'?"

If the answer is "Yes", then Sauron is alive. If it is "No", then Sauron is dead.

Explinition:

If the answer is "Yes", consider the following possibilities:

a) If he is a sane elf, then Sauron is alive.

b) If he is an insane elf, then the two statements are not equivalent. Since he is neither a sane elf nor an insane human, the first statement is false. Therefore, Sauron must be alive.

c) If he is a sane man, then the two statements are not equivalent. As in (b), Sauron must be alive.

d) If he is an insane man, then the two statements are equivalent, so Sauron is alive.

If the answer is "No", consider the following possibilities:

a) If he is a sane elf, then the two statements are not equivalent. Since he is a sane elf, the first statement is true. So Sauron must be dead.

b) If he is an insane elf, then the two statements are equivalent. Since he is neither a sane elf nor an insane human, the first statement is false. Therefore, Sauron is dead.

c) If he is a sane man, then the two statements are equivalent. Again, this means that Sauron is dead.

d) If he is an insane man, then the two statements are not equivalent. Again, this means that Sauron is dead.

Question 1

S S Hanuram Vikash 22272 <hanuram22@iiserb.ac.in>

Bhumika Upadhyay 22080 <bhumika22@iiserb.ac.in>

Question 2

Bhumika Upadhyay 22080 <bhumika22@iiserb.ac.in> solved 2A and 2B correctly.

Question 3

Almost to the right answer.

Hrishit Gupta 22152 <hrishit22@iiserb.ac.in>

Ujjivan Mitra 230405 <ujjivan23@iiserb.ac.in>

Correct answer

Soumyadeep Gope 22329 <soumyadeepg22@iiserb.ac.in>