## Cheryl's Birthday spinoff

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Wildcard
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### Cheryl's Birthday spinoff

Cheryl hides a date, this time one complete with day, month and year. She tells the day to Danny, the month to Mark and the year to Yolanda. This much (this first paragraph) is common knowledge, along with the following list of possible dates which she provides:

7 August 2009
14 April 2010
23 April 2010
17 August 2010
7 December 2010
14 April 2011
17 August 2011
7 December 2011
17 January 2012
14 April 2013
5 October 2013
7 November 2013
23 February 2014
23 April 2014
5 August 2014
5 January 2015

The following conversation then takes place:

Yolanda: I don't know the date.
Danny: I don't know the whole date, but I do know the month.
Mark: Now that you've said that, I know the day, but not the year.
Yolanda: I now know the entire date.
Danny: In that case Mark and I know the date now also!

What is the date?

(Note: I wrote this one myself. I started by writing the list of clues and then worked backwards to generate a list. It wasn't easy...but I've triple checked and I'm certain that I did it completely correctly such there is a unique and logically deducible solution. Enjoy!

(Extra note: These puzzles are a heckuva lot harder to write than to solve....)
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Moonbeam
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### Re: Cheryl's Birthday spinoff

Seeing how no one has had a go yet, I've had a quick look and get:

Spoiler:
14th April 2013 ??

.... it was only a quick go, so I could be wrong >.<

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### Re: Cheryl's Birthday spinoff

Spoiler:
I think this is solvable without Mark though. After Danny's first statement, everyone knows it's 14 April or 17 August, so Mark's statement doesn't add anything and Yolanda knows the date without any other info.

Perhaps this works: get rid of 23 February 2014, change 17 August 2011 to 17 August 2012, and change Yolanda's last statement to "I didn't know the date until Mark said anything, but now I know the date". Then after Danny's first statement everyone knows it's either 14 April, 17 August, OR 23 April. Mark's statement points to it being 17 August, and it must not have been 17 Auguest 2012 if Yolanda didn't figure it out before Mark's statement, so it must be 17 August 2010.

That might not be right either, and maybe I missed something and the original problem is better. These problems are hard!

We played a forum game a couple years ago that similar to this but a bit twistier.
Last edited by Adam H on Fri Apr 24, 2015 6:30 pm UTC, edited 1 time in total.

rmsgrey
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### Re: Cheryl's Birthday spinoff

They're easier to write if you work backwards.

If the goal is January 1st, and the guy who knows the day is first to know the full truth, then you need another date in January, say, January 2nd, which will be left after the previous deductions. For that to be useful to the guy who knows the month, you need another 2nd to rule out, say February 2nd.

So your target is January 1st, January 2nd, February 2nd, but you need the guy with the months to not know yet, so add Jan 3rd and Feb 3rd. But for the guy with the days not to have known yet, you need to have just ruled out another month with the 1st as an option, so add in March 1st and March 4th.

So, with those 7 dates, the guy with the month knows nothing more until the guy with the date says he doesn't know, ruling out the 4th. The guy with the month still doesn't know, eliminating March, letting the guy with the date know it's Jan 1st, and the fact he knows tells the guy with the month that it's the 1st.

You can then add on another layer - say April 4th - the guy with the month still doesn't know, but the guy with the date can only eliminate April once he knows that...

EricH
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### Re: Cheryl's Birthday spinoff

Working through the logic:

Spoiler:
Yolanda knows the year, but not the date. If the year were 2009, 2012, or 2015, there would only be one date possible, so her first statement would be false. Therefore, the year is 2010, 2011, 2013, or 2014.

Danny knows the day of the month, which must be 5, 7, 14, 17, or 23, and now knows the year is one of the four above.
If it were 5, he wouldn't be able to tell whether it was 5 October 2013, or 5 August 2014, so it isn't 5.
If it were 7, he wouldn't be able to tell whether it was 7 December 2010, 7 December 2011, or 7 November 2013, so it isn't 7.
If it were 14, it could be 14 April 2010, 14 April 2011, or 14 April 2013. In any case, he would know that it's April.
If it were 17, it could be 17 August 2010, or 17 August 2011. Either way, he would know that it's August.
If it were 23, he wouldn't be able to tell whether it was 23 April 2010, 23 February 2014, or 23 April 2014, so it isn't 23.

Mark knows the month, which must be April or August. If it's April, he knows it's 14 April, but it could be 2010, 2011, or 2013.
If it's August, he knows it's 17 August, but it could be 2010 or 2011. So his statement doesn't tell us anything more than Danny's did.

Yolanda knows whether it's 2010, 2011, or 2013--but now, she knows which date it is. If the year were 2010 or 2011, she still wouldn't be certain whether it was 14 April or 17 August, so it must be 2013, and therefore 14 April 2013.

Obviously, if we can figure it out without knowing day, month, or year, then Danny and Mark can do the same, so Danny's last statement doesn't tell us anything new, either.

Challenge: Make a version where every statement is required for a solution, and no one person's statement reveals the answer to more than one of the others simultaneously. (In other words, if we reach a point where Yolanda says "Now I know the entire date", then either Mark or Danny still wouldn't know, and we the audience have to wait until all three have figured it out, before we have enough info to solve the puzzle.)
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Wildcard
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### Re: Cheryl's Birthday spinoff

Spoiler:
I think this is solvable without Mark though. After Danny's first statement, everyone knows it's 14 April or 17 August, so Mark's statement doesn't add anything and Yolanda knows the date without any other info.

Perhaps this works: get rid of 23 February 2014, change 17 August 2011 to 17 August 2012, and change Yolanda's last statement to "I didn't know the date until Mark said anything, but now I know the date". Then after Danny's first statement everyone knows it's either 14 April, 17 August, OR 23 April. Mark's statement points to it being 17 August, and it must not have been 17 Auguest 2012 if Yolanda didn't figure it out before Mark's statement, so it must be 17 August 2010.

That might not be right either, and maybe I missed something and the original problem is better. These problems are hard!

We played a forum game a couple years ago that similar to this but a bit twistier.
I think you're right about it being solvable without Mark. But I checked your proposed changes, and
Spoiler:
I get that with the list changed as you specify, Mark's statement is logically impossible. Here is the changed list:

7 August 2009
14 April 2010
23 April 2010
17 August 2010
7 December 2010
14 April 2011
7 December 2011
17 January 2012
17 August 2012
14 April 2013
5 October 2013
7 November 2013
23 April 2014
5 August 2014
5 January 2015

After Yolanda's statement, we have the following:

7 August 2009
14 April 2010
23 April 2010
17 August 2010
7 December 2010
14 April 2011
7 December 2011
17 January 2012
17 August 2012
14 April 2013
5 October 2013
7 November 2013
23 April 2014
5 August 2014
5 January 2015

After Danny's statement, we have the following:

7 August 2009
14 April 2010
23 April 2010
17 August 2010
7 December 2010
14 April 2011
7 December 2011
17 January 2012
17 August 2012
14 April 2013
5 October 2013
7 November 2013
23 April 2014
5 August 2014
5 January 2015

At this point it is common knowledge that the date is in April. So if Mark could figure out the day, then we as bystanders should be able to also. But we still have possibilities of it being 14 April or 23 April. So Mark is lying (or mistaken).

You're sure right about these problems being hard, though!
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Wildcard
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### Re: Cheryl's Birthday spinoff

Possibly this post should go in its own thread. But it's not properly a logic puzzle yet, it's an attempt to write one.

I have another list of clues that I wrote, and I started trying to write a list of dates which makes a solvable puzzle. This one is just completely absurdly hard. But I REALLY want to either (a) come up with a list of dates that makes a solvable puzzle or (b) logically prove that such a list is impossible.

Here is the clue list. The setup is the same as in the puzzle I posted originally, except that there is also a bystander, Billy, who gets no information beyond the setup of the puzzle.

Yolanda: I don't know the date.
Danny: Shall I tell you whether I know the date or not?
Yolanda: Even if you told me that I would not know the date.
Mark: I know the date.
Yolanda: In that case, Danny, if you tell me whether you know the date or not, I will know the date!
Danny: I know the date.
Yolanda: Now I know the date.
Billy: I don't know the date.
Yolanda: I knew that Danny's answer (of whether or not he knew the date) wouldn't allow you to figure out the date, even before he answered.
Billy: In that case I know the date.

There are some statements here that are logically null (i.e. make no difference to anyone's knowledge). These are
Spoiler:
Danny's first question (of course) and Yolanda's statement that she knows the date, and both statements by Billy (since we as the puzzle readers have exactly the knowledge that Billy has).

I just get utterly baffled every time I try to deduce anything which must necessarily be true about the list of dates. I suspect that there is a contradiction in Yolanda's last statement and the fact that it allows Billy (and us) to deduce the date, but I can't seem to pin it down....
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rmsgrey
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### Re: Cheryl's Birthday spinoff

Wildcard wrote:I suspect that there is a contradiction in Yolanda's last statement and the fact that it allows Billy (and us) to deduce the date, but I can't seem to pin it down....

There is.

Yolanda's secret information (the year) has no bearing on what she can deduce about Billy's knowledge - any deductions anyone makes about Billy's knowledge can be made entirely using public information because everything he knows is public information. If Yolande knew that Billy wouldn't be able to tell the date from Danny's answer, then so did Billy, so her telling him she already knew that doesn't add anything to his knowledge.

In the original problem, A's secret information told him something about B's secret information, which enabled A to deduce things which weren't deducible by bystanders, so A sharing that deduction provided information about A's secret information.

Wildcard
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### Re: Cheryl's Birthday spinoff

Spoiler:
I don't think that's quite it. It isn't about what she deduced about HIS knowledge, it's what she deduced about what his knowledge WOULD BE after a piece of information (which she didn't know specifically) was added to his knowledge. The fact that she could predict his knowledge based only on the information she had, means that her information uniquely specified something about the date....

Basically, because she knows the year, she can eliminate more more possibilities than Billy. But I still think there is a contradiction....

Yolanda knows that after Danny states whether he knows the date or not, she will know the answer -> she has narrowed it down to two possible dates, one of which would allow Danny to know the entire date at that point in the puzzle, and one of which would NOT allow Danny to know the entire date. (I'm pretty sure of my logic here...I don't think she can have more than two dates she hasn't eliminated, by the time she asks Danny for whether he knows the date or not.)

If Danny doesn't know the entire date, then of course Billy can't know the entire date either because Billy has less info than Danny.

If Danny DOES know the date (at the point when Yolanda realizes that [finding out whether Danny knows the date or not] will allow her to determine the date), then there is some possibility that Billy could know the date also, at the same time as Yolanda learns the date. But if Yolanda has narrowed it down to only two possible dates, one of which Danny would know and one of which he wouldn't, then she would know whether the info that Danny knows the date would sufficiently narrow down the common knowledge possibilities to ONE date or not...um, I think my brain just broke.

Try it from the other side: if Yolanda's final statement allows Billy to narrow down anything, it means that he was considering at least two possible dates: one of which would have allowed Yolanda to predict that Billy would know the answer after Danny's input, and one of which wouldn't have allowed that, but both of which would have allowed Yolanda to predict that SHE would know the date after Danny's input. That means that Yolanda's final statement would eliminate the possibility that it was the one that wouldn't have let her predict ahead of time that Billy wouldn't know the answer after Danny's input.
Lengthy monologue of logic spoilered for length. Conclusion below.

rmsgrey wrote:Yolanda's secret information (the year) has no bearing on what she can deduce about Billy's knowledge
is true, but
No one so far wrote:Yolanda's secret information (the year) could possibly have a bearing on what she can deduce about Billy's FUTURE state of knowledge, after Danny answers a specified question the answer to which is not public information.
...Oh. I finally got it—the key to the above statement is that Yolanda must be able to determine what Danny's answer will be, specifically, in order to deduce what Billy's knowledge state will be after the answer is given, UNLESS it is possible with public information to deduce what Billy's knowledge state will be after the answer is given. Since it's a given in the puzzle that she DOESN'T know what Danny's answer will be, then although it would still be possible for her to deduce Billy's future knowledge state for each of Danny's possible answers, her secret information couldn't have any bearing on that deduction and therefore stating that she had made that deduction couldn't have clarified anything for Billy.
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rmsgrey
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### Re: Cheryl's Birthday spinoff

Wildcard wrote:
rmsgrey wrote:Yolanda's secret information (the year) has no bearing on what she can deduce about Billy's knowledge
is true, but
No one so far wrote:Yolanda's secret information (the year) could possibly have a bearing on what she can deduce about Billy's FUTURE state of knowledge, after Danny answers a specified question the answer to which is not public information.
...Oh. I finally got it—the key to the above statement is that Yolanda must be able to determine what Danny's answer will be, specifically, in order to deduce what Billy's knowledge state will be after the answer is given, UNLESS it is possible with public information to deduce what Billy's knowledge state will be after the answer is given. Since it's a given in the puzzle that she DOESN'T know what Danny's answer will be, then although it would still be possible for her to deduce Billy's future knowledge state for each of Danny's possible answers, her secret information couldn't have any bearing on that deduction and therefore stating that she had made that deduction couldn't have clarified anything for Billy.

Exactly. This sort of thing gets tricky because you're talking about what people know about each other's knowledge, and several people have secrets which (may) give them partial knowledge of other people's secrets.