bonanova

Can you gain as much by switching back? Using the same reasoning? There is a paradox: A gain for switching can be anticipated. Yet, there is a preferred envelope, and if we initially chose it we should not switch.

Impressive. I think 10x10 is the largest known square. How large will your program go?

Is there no end to circumstances that ask whether we should switch? This one might take some thought. Your rich uncle places a sum of money into an envelope, twice that amount into a second envelope, places them both into a black bag and invites you to draw one out of the bag. You open the envelope that you draw and find $1000. You now know the envelope in the bag contains either $500 or $2000. Your uncle gives you the option to take the other envelope in exchange for the one you drew. Calculate the expected gain that comes from switching.

Do they take turns?

Let's take it up a notch to seven letters. Can M E R G E R S be squared?

Nice.

By accepting, Davey promised a truthful answer.

jawk got it first, but see witzar's post for the analysis. Bushindo has to account for the 11 hours and 59 minutes consumed discussing a two-digit number

Both d_m and Ms got it right. I marked Ms's post, in order to point readers to the logic he provided.

Both BobbyGo and dark_magician said they would take the challenge. Here is the question to which they promised a truthful answer: Will you either answer no to this question or pay me two million dollars? This question courtesy of Raymond Smullyan.

Yes. I meant integer. And both answers are correct.

Alex just offered Davey $1,000,000 to answer a yes/no question truthfully. Would you advise Davey to accept the offer? Why or why not?

From the closed interval [1,1000] I just chose a number; I invite you to do the same. Now we ask: what is the probability that your number is higher?

The wise man on the mountain knew the secret of life. The seeker was granted a 12-hour audience; all he had to do was climb. He began the climb precisely at 8:00 one morning, climbing at the rate of 1.5 mph. After his meeting with the seer, he immediately descended using the same trail, at 4.5 mph, and he reached the bottom at noon the next day. And now what question shall we ask? A. What is the meaning of life told to the seeker? B. How long is the trail? Question B.

Alex was charged with robbing a bank. At trial Ian testified that Alex had committed more than a dozen robberies during his lifetime. Jamie denied Ian's statement. Davie then opined that while Alex may not have committed a dozen robberies, he had in fact committed at least one robbery. It was later determined that only one of the witnesses had told the truth. Was Alex guilty?

Barbara, Kathryn and Peter are siblings. Either Peter or Barbara is the oldest of the three. Either Kathryn is the oldest or Peter is the youngest. In what order with the three born?

There is not enough information to solve. Proof: Draw BC of length 6.5. From C, draw a ray R1 inclined 15o. From B, draw a ray R2 inclined at some angle in the first quadrant. Where R2 intersects R1 (call it point D) proceed one unit further and place point A. Draw AC. Now you have the figure given in post #2. Angle B is any first-quadrant angle. Angle A depends on angle B; it has no discoverable value.

So the 4/5 liar could not respond like this? TLLLL LLLLT ..... ..... That is, he could never lie on 5 consequetive responses? That makes it much easier. He could. What is meant in the op is that for every set of five questions the 4/5 liar will lie four times (with 4 lies at the end possibly). if asked another set of five questions the liar will again lie four out of five times (with 4 lies at the beginning this time). So: if asked 5 questions T L L L L (possibly) next 5 questions L L L L T (possibly) so T L L L L L L L L T = 8L / 10total which reduces to 4/5 So it's four lies for question 1-5. Four more lies for questions 6-10 and so forth. Not quite what the OP says. If it were four lies for every "five" questions then the lies, after the first truth, would always come in groups of exactly four. It could not be T LLLLL LLLT because questions 2-6 comprise "five" questions that do not contain a truth. OK now it's hard again.

So the 4/5 liar could not respond like this? TLLLL LLLLT ..... ..... That is, he could never lie on 5 consequetive responses? That makes it much easier.

Agree.