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Suppose a man only tells the truth three hours a day. It is the same every week and it is known that the pattern of the truth hours are TFTFT. (which means he may tell the truth at midnight and 2am and 4am, 7am and 11am and 1pm, and so on). On the other 21 hours, he lies. All times between 7pm and 6am inclusive are considered night, 7am to 11am are morning, noon to 4pm are afternoon, and 5pm to 6pm are evening.

He gives the following statements on seven consecutive hours.

I always lie between 3pm and 8pm.

It's morning now.

I lie between 8am and noon and/or the whole night.

I lie the whole afternoon.

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

It's somewhere between 9am and 4pm now.

I lie on 1am, 4pm, 8pm and/or midnight.

Questions:

What are the chances that he lies the whole evening?

What are the chances that none of these statements were spoken in the morning?

What are the chances that he tells at most one true statement between 3pm and 7pm?

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  • $\begingroup$ "7am to 11pm are morning", should that be 11am? $\endgroup$
    – hexomino
    Jun 23, 2022 at 14:10
  • $\begingroup$ In the last statement, there is a difference between "and" and "or", which do you mean to use? $\endgroup$
    – hexomino
    Jun 23, 2022 at 14:28
  • $\begingroup$ Are you ever going to accept answers or address the open questions on the previous puzzles you've posted here on the site or do you just post puzzles and then ignore them? $\endgroup$
    – SQLnoob
    Jun 23, 2022 at 16:04
  • $\begingroup$ "7am and 11am and 1pm" - did you mean "9am and 11am and 1pm"? (i.e. is "TFTFT" intended to mean that the second truthful hour is exactly two hours after the first, and the third is exactly two hours after the second?) $\endgroup$
    – Ed Murphy
    Jun 24, 2022 at 7:48

1 Answer 1

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First note that

the 7th statement is always true, regardless of when it's spoken. Therefore statements 1, 2, 4 and 6 must all be lies.

In trying to figure out when the 7th statement is actually being made, statement 4 being a lie means that he must tell the truth at some point in the afternoon, which rules out an 11-hour window; and statement 6 being a lie rules out another eight hour window.

At this point we've narrowed down the time of the 7th statement to 8AM, 9AM, 6PM, 7PM or 8PM. However, it can't have been made at 8 or 9 AM, since that would make statement 1 true. And it can't be 6PM, as that would cause a contradiction with statement 5.

So the 7th statement is made at 7 or 8 PM, and in either case this is the third of the set of three true statements in a 5-hour period in order to fulfill the requirement that there is a true statement spoken in the afternoon.

What are the chances that he lies the whole evening?

0%. He either tells the truth at 5PM or 6PM.

What are the chances that none of these statements were spoken in the morning?

100%. They were made between 1-7 PM or 2-8 PM.

What are the chances that he tells at most one true statement between 3pm and 7pm?

0%. He either tells the truth at 3, 5, and 7 PM or 4, 6, and 8 PM.

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