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A man is only capable of telling the truth four hours a day. They are either TTFTT or TFTTT (equiprobable). For the other twenty hours, he lies. All hours between 7pm and 6am are night; 7am and 11am are morning; noon and 4pm are afternoon; and 5pm and 6pm are evening.

He gives these statements in seven consecutive hours:

  1. I lie between 4pm and 8pm.

  2. It's afternoon now.

  3. I tell at most two true statements at night.

  4. Evening will come in less than three hours.

  5. It's somewhere between 4pm and midnight now.

  6. lie at 9pm, 1am, and 7am.

  7. Either I lie the whole evening, or it's morning now, but not both.

Questions:

  1. What are the chances he tells the truth less than twice in the morning?

  2. What are the chances that less than two of these statements were spoken in the evening?

  3. What are the chances he tells the truth less than four times at night?

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  • $\begingroup$ Can you clarify whether the statements have to be made in the order listed, or can be in any order as long as they are in seven consecutive hours? $\endgroup$
    – fljx
    Jul 25 at 8:53

1 Answer 1

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UPDATED ANSWERS

First off, here is what’s NOT possible:

The first statement cannot have been made at 11am. This would make statements 2 and 3 true, and statements 4 and 5 false, which would then make statement 1 true. So the true statements would be 9am, 11am, 12pm, 1pm. But this would make statement 6 true, which is a contradiction.

The first statement cannot have been made at 12pm or 1pm. This would make statements 2 through 5 all true, which is a contradiction.

The first statement cannot have been made between 4-8pm. This would make statements 2 and 4 lies, and statement 5 true. This, in turn, would make statements 3 and 1 a lie. and the subsequent alignment of true statements would cause a contradiction with statement 6 or 7.

The first statement cannot have been made at 9pm. This would make statements 2, 4, and 5 lies, which would make statements 3 and 1 lies. This would mean the true statements would be made sometime between 2-7 am, but this would then make statement 1 true, which is a contradiction.

Of the remaining cases:

If the first statement is made at 2 or 3 pm, then statements 2, 3, 5 and 6 are true. (2 cases)

If the first statement is made at 10 pm, then statements 2, 4 and 5 are lies. Then statement 3 must be a lie, so 3+ true statements are made at night, which is only possible if they are at 5pm and 7-9pm. (1 case)

By similar logic, if the first statement is made at 11pm, 12am, or 6-10am, then the true statements are made at 5pm and 7-9pm; 6-7 and 9-10pm; or 6 pm and 8-10pm. (3 cases each * 7 start points = 21 cases)

If the first statement is made from 1-5am, then the true statements are made between 7pm and the time of the first statement. These are harder to enumerate, but can be seen in the following image.

The image below illustrates all of the possibilities:

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Each row is a possible case. The blue-colored boxes are hours when true statements are made, and pink are lies. The outlined box is the seven-hour window during which the statements are made.

There are

59 rows corresponding to 59 possible arrangements of the statements (in terms of when they were spoken, and when in the day the true statements could have been made). So answering the questions again, in this context, I have:

What are the chances he tells the truth less than twice in the morning?

100%. As you can see in the illustration, there is no case where any true statements are made in the morning.

What are the chances that less than two of these statements were spoken in the evening?

57 / 59. As seen in the illustration, only the top two rows span the evening hours. In all other cases the statements are spoken at a different time of day.

What are the chances he tells the truth less than four times at night?

24 / 59. As seen in the illustration, the top 24 rows have at least one true statement made at a time outside of the night hours.

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