# Birthday Logic Question **URL:** \ **Category:** Factual Questions\ **Created:** [April 14, 2015, 5:31am UTC](https://boards.straightdope.com/t/birthday-logic-question/717622 "2015-04-14T05:31:35Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![Cheesesteak](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cheesesteak/32/3694_2.png) [@Cheesesteak](https://boards.straightdope.com/u/Cheesesteak)\ **Post date:** [April 14, 2015, 3:16pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/41 "2015-04-14T15:16:42Z")
> [@jezzaOZ](#): > > B still has multiple months with single dates he can choose from. No, he doesn’t. A says he KNOWS that B CANNOT KNOW the date. If B was given a 19, he can say “you’re wrong, I do know the date, it is May 19, because I was told 19, and that’s the only day with 19” Since we’re told that B cannot do this, May (and June) are right out. B says that he now knows the date. We’re onto July and August only, and dates 15, 16 and 17 can allow B to know the date for certain when you’re looking at those two months. If he was given 14, he couldn’t know the date. A also says that he now knows the date, which means out of July 16, August 15 and August 17, he can nail down the date for certain, which means he must have been given July. Thus the date is July 16 ---
**Author:** ![Telemark](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/telemark/32/372_2.png) [@Telemark](https://boards.straightdope.com/u/Telemark)\ **Post date:** [April 14, 2015, 3:21pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/42 "2015-04-14T15:21:53Z")
> [@Bricker](#): > > I feel it is these facts which are confusing the OP. No, he claims to accept these conditions, he just doesn’t agree with the logic. He’s wrong, but we’re working on fixing that. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [April 14, 2015, 3:24pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/43 "2015-04-14T15:24:36Z")
OK, rewritten to keep in the realm of reality: 10♥ J♥ A♥ Q♦ K♦ 9♠ J♠ 9♣ 10♣ Q♣ I guess that’s why I originally was thinking in terms of suits, as I wanted to associate the colors (to help the visualization) with the suits, not the numbers, but either way works. You can follow the logic the same way. I’m sure there’s probably even another, simpler way, to visualize it, but this is how my brain works. (Also, because our first player was given the month, I was mapping month-day to rank-suit, the way cards are conventionally notated.) ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [April 14, 2015, 3:29pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/44 "2015-04-14T15:29:13Z")
I got exactly the same solution as Richard Pearse and he explained it as succinctly as I could so I won’t repeat it. ---
**Author:** ![nolonger\_lurking](https://avatars.discourse-cdn.com/v4/letter/n/41988e/32.png) [@nolonger\_lurking](https://boards.straightdope.com/u/nolonger_lurking)\ **Post date:** [April 14, 2015, 3:31pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/45 "2015-04-14T15:31:19Z")
> [@jezzaOZ](#): > > I fully understand the initial conditions and state of knowledge of the parties. What I don’t accept is the logical leaps involved to get one of many proffered solutions. I state the problem is insoluble and ask that you come up with a well reasoned argument to prove otherwise. Any logic puzzle of this sort can be brute-forced by testing each of the options against all the statements. While that is often unreasonable by hand, it isn’t so bad in this case. Are you saying that July 16th fails any of the statements, or are you saying that one of the other dates passes? If so, what other date works? ---
**Author:** ![Czarcasm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/czarcasm/32/4050_2.png) [@Czarcasm](https://boards.straightdope.com/u/Czarcasm)\ **Post date:** [April 14, 2015, 3:31pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/46 "2015-04-14T15:31:53Z")
I’m not buying it. High school students in Singapore are given logic problems that use names like “Cheryl”, “Bernard” and “Albert”? ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [April 14, 2015, 3:35pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/47 "2015-04-14T15:35:08Z")
Or perhaps this way, to keep the order of information consistent (that is, the first part is what Albert knows, the second part is what Bernard knows): ❤10 ❤J ❤A ♦Q ♦K ♠9 ♠J ♣9 ♣10 ♣Q I’ll shut up now. ---
**Author:** ![Omar\_Little](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/omar_little/32/269_2.png) [@Omar\_Little](https://boards.straightdope.com/u/Omar_Little)\ **Post date:** [April 14, 2015, 3:39pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/48 "2015-04-14T15:39:54Z")
> [@Czarcasm](#): > > I’m not buying it. > > High school students in Singapore are given logic problems that use names like “Cheryl”, “Bernard” and “Albert”? English is the official language of the country. ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [April 14, 2015, 3:42pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/49 "2015-04-14T15:42:13Z")
> [@Nametag](#): > > Okay, let’s do it the long way. > > If Cheryl had whispered “19” to Bernard, then he would instantly know that her birthday is May 19th, because May 19th is the only “19” on the list. Similarly, if she had whispered “18,” Bernard would know that her birthday is June 18th, because June 18th is the only “18” on the list. These are the only unique numbers in the list - if she tells Bernard “14,” “15,” “16,” or “17,” then he can’t know which month she was born in, because each of those numbers appears in at least two months. > > Now for Albert. If Cheryl whispers “May” to Albert, it raises the possibility that she whispered “19” to Bernard - in which case he would know her birthday is May 19th. Similarly, if she whispers “June,” it raises the possibility that she whispered “18” to Bernard - in which case he would know that her birthday is June 18th. So if Albert knows that Bernard doesn’t know her birthday, it can only be because those possibilities were eliminated entirely just by knowing the month - in other words, she didn’t tell Albert “May” or “June.” That leaves July and August. > > Back to Bernard. He, being almost as clever as a Doper, has worked out all of the above from what Albert said. So now let’s game out the other dates: > > If Cheryl told Bernard “14,” he still can’t tell what month she was born in, because July 14th and August 14th are both on the list. > > If Cheryl told Bernard “15,” then he now knows she was born on August 15th, because that’s the only 15 on the list. > > If Cheryl told Bernard “16,” then he now knows she was born on July 16th, because that’s the only 16 on the list. > > If Cheryl told Bernard “17,” then he now knows she was born on August 17th, because that’s the only 17 on the list. > > So if Bernard declares confidently that he now knows Cheryl’s birthday, it’s because it’s either August 15th, July 16th, or August 17th. I agree with all of this. > [@Nametag](#): > > Back to Albert. He, being almost as clever as Bernard, has also worked out all of the above possibilities, and confidently declares that he, too, now knows Cheryl’s birthday. If Cheryl had told him “August,” he wouldn’t be able to do that, because there are two possible dates in August, the 15th and the 17th. So if Albert knows Cheryl’s birthday, it can only be because she told him “July,” thereby eliminating the dates in August. This leaves the one remaining date - which, huzzah, is in July - July 16th. But I don’t agree with this. It’s true that July 16th is the only date Albert would be able to deduce from the information he has - but he has no reason to assume he has enough information to deduce the correct date. It’s possible that Cheryl likes Bernard better than Albert so she carefully gave them information which would reveal her birthdate to Bernard but not reveal it to Albert. The solution given would only work if Cheryl had said that she had given the boys enough information for both of them to figure out her birthday. Then Albert would know the problem was solvable and July 16 was the only possible solution. ---
**Author:** ![Czarcasm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/czarcasm/32/4050_2.png) [@Czarcasm](https://boards.straightdope.com/u/Czarcasm)\ **Post date:** [April 14, 2015, 3:42pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/50 "2015-04-14T15:42:51Z")
> [@Omar\_Little](#): > > English is the official language of the country. I knew that…but are those common names in Singapore? ---
**Author:** ![don\_t\_ask](https://avatars.discourse-cdn.com/v4/letter/d/e68b1a/32.png) [@don\_t\_ask](https://boards.straightdope.com/u/don_t_ask)\ **Post date:** [April 14, 2015, 3:43pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/51 "2015-04-14T15:43:33Z")
To the OP. Let’s say you are playing Bernard’s role. Cheryl tells you the number. If it is 18 or 19 do you now know her birthday? So how can Albert be sure that you don’t **already** know her birthday? ---
**Author:** ![Omar\_Little](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/omar_little/32/269_2.png) [@Omar\_Little](https://boards.straightdope.com/u/Omar_Little)\ **Post date:** [April 14, 2015, 3:44pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/52 "2015-04-14T15:44:43Z")
> [@Little\_Nemo](#): > > It’s true that July 16th is the only date Albert would be able to deduce fro the information he has - but he has no reason to assume he has enough information to deduce the correct date. Yes A does. He has the month, with which he is able to deduce the day, since B says they know both the day and month, now. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [April 14, 2015, 3:46pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/53 "2015-04-14T15:46:23Z")
> [@Little\_Nemo](#): > > I agree with all of this. > > But I don’t agree with this. > > It’s true that July 16th is the only date Albert would be able to deduce fro the information he has - but he has no reason to assume he has enough information to deduce the correct date. > > It’s possible that Cheryl likes Bernard better than Albert so she carefully gave them information which would reveal her birthdate to Bernard but not reveal it to Albert. In that case, Albert would have been given the month August, which would not make his last statement possible. July is the only month he could have been given for that statement to be true, so, for the purposes of this puzzle, that’s the only date that is correct, given all the the statements are true. ---
**Author:** ![flight](https://avatars.discourse-cdn.com/v4/letter/f/bbce88/32.png) [@flight](https://boards.straightdope.com/u/flight)\ **Post date:** [April 14, 2015, 3:46pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/54 "2015-04-14T15:46:33Z")
> [@jezzaOZ](#): > > The question doesn’t mention whispers… He used whisper to describe when Cheryl told the other two their Month or Day. Seriously, reread **Nametag** ’s post. He gives a perfect and logical breakdown of the solution. This is the supported logic you have been asking for. ---
**Author:** ![chrisk](https://avatars.discourse-cdn.com/v4/letter/c/6de8d8/32.png) [@chrisk](https://boards.straightdope.com/u/chrisk)\ **Post date:** [April 14, 2015, 3:47pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/55 "2015-04-14T15:47:05Z")
> [@nolonger\_lurking](#): > > Any logic puzzle of this sort can be brute-forced by testing each of the options against all the statements. While that is often unreasonable by hand, it isn’t so bad in this case. Are you saying that July 16th fails any of the statements, or are you saying that one of the other dates passes? If so, what other date works? I’m confused by this. How do you pass or fail these kind of statements, which call for reasoning on the part of the characters in the puzzle? If a statement is true, but depends on information that character doesn’t have, would it “pass”? > [@Little\_Nemo](#): > > I agree with all of this. > > But I don’t agree with this. > > It’s true that July 16th is the only date Albert would be able to deduce fro the information he has - but he has no reason to assume he has enough information to deduce the correct date. > > It’s possible that Cheryl likes Bernard better than Albert so she carefully gave them information which would reveal her birthdate to Bernard but not reveal it to Albert. But in the puzzle, it’s given that Albert believes he has deduced the correct date and proclaims it at the end. Are you leaving open the possibility that he made a mistake? 😉 ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [April 14, 2015, 3:50pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/56 "2015-04-14T15:50:51Z")
> [@Omar\_Little](#): > > Yes A does. He has the month, with which he is able to deduce the day, since B says they know both the day and month, now. Not quite. B only says he himself knows the date. He doesn’t say they both know the date. (But he could come to that conclusion. He just doesn’t give that info in the puzzle as written.) ETA: unless you are just using “they” reflexively in a gender-neutral sense to refer to the presumably male Bernard. ---
**Author:** ![Kimble](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kimble/32/19185_2.png) [@Kimble](https://boards.straightdope.com/u/Kimble)\ **Post date:** [April 14, 2015, 3:51pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/57 "2015-04-14T15:51:24Z")
What this thread needs is another participant! > [@jezzaOZ](#): > > I certainly agree with your initial conditions. What I don’t agree with is that you can discard months with unique dates. You have to remember that the first statement is made on purely public knowledge. A can only know that B has a number of dates and B has no information about what A knows. When A makes their pronoucement all B can learn is that A thinks he has no idea _snip_ Not “thinks” – "knows."A knows that B doesn’t know. (You do agree with **chrisk** ’s “…implicit assumption that A and B each know that the other will make correct and logical deductions about the information available,” right?) B can now think about what A said and figure out how A knows that B doesn’t know – by brute force if necessary. What B discovers is that A must have been told either July or August by C. Everything else follows in a similar way. ---
**Author:** ![Doug\_K](https://avatars.discourse-cdn.com/v4/letter/d/b5ac83/32.png) [@Doug\_K](https://boards.straightdope.com/u/Doug_K)\ **Post date:** [April 14, 2015, 4:57pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/58 "2015-04-14T16:57:02Z")
> [@bienville](#): > > No one is asking the obvious question: Why wouldn’t Albert and Bernard at this point immediately choose to _not_ remain friends with Cheryl? Seriously. Cheryl is being ridiculous. If she’s going to be this difficult having just begun the friendship, she’s only going to get worse as time goes one. > Albert and Bernard should have just turned and walked away. They just met. It’s none of their damn business what her birthday is. I don’t know what Albert and Bernard are up to, but normal people don’t demand to know the birthdates of people they just met. Cheryl should have just turned and walked away. And changed all her passwords to be safe. ---
**Author:** ![nolonger\_lurking](https://avatars.discourse-cdn.com/v4/letter/n/41988e/32.png) [@nolonger\_lurking](https://boards.straightdope.com/u/nolonger_lurking)\ **Post date:** [April 14, 2015, 5:08pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/59 "2015-04-14T17:08:51Z")
> [@chrisk](#): > > I’m confused by this. How do you pass or fail these kind of statements, which call for reasoning on the part of the characters in the puzzle? If a statement is true, but depends on information that character doesn’t have, would it “pass”? July has 2 dates - 14 and 16, so Albert doesn’t know the date. Because August 14th and May 16 are also options, he knows that no matter which of the two days is correct, Bernard can’t guess the month. Compare this to Albert hearing that the month is May- the possible birthdays are May 15,16,19. If the birthday is May 15 or 16, Bernard doesn’t know the birthday, because August 15 and July 16 are possibilities to Bernard. But if the Birthday is May 19, then Bernard knows the day is 19- that is all he needs to know the whole birthday. This means that birthdays in May are not compatible with statement 1. The same analysis is true of birthdays in June. Bernard is a great logician and he knows this- it gives him information about the Month given to Albert. Bernard knows from the beginning that the day is the 16th. Combined with the information gleaned from Albert’s statement (the month cannot be May or June), there is only one birthday in July or August that is on the 16th- Bernard now knows that the birthday is July 16th. Compare this to him hearing the other days: 14: the birthday could be July 14 or August 14- not enough information. This contradicts Bernard’s statement that he now knows the birthday 15: birthday would be August 15, but we will rule that out later 17: birthday would be August 17, but we will rule that out later 18,19:These days are only in May and June, so they conflict with Albert’s first statement. Albert knows that the month is July. If the birthday is July 14, then Bernard has insufficient information even after Albert speaks. This contradicts Bernard’s statement. Thus Albert knows that that the birthday is July 16. For each Bday: May 15 - Fails statement 1- Albert can’t be confident that Bernard doesn’t know the whole birthday May 16 - Fails statement 1- Albert can’t be confident that Bernard doesn’t know the whole birthday May 19 - Fails statement 1- Albert can’t be confident that Bernard doesn’t know the whole birthday June 17 - Fails statement 1- Albert can’t be confident that Bernard doesn’t know the whole birthday June 18 - Fails statement 1- Albert can’t be confident that Bernard doesn’t know the whole birthday July 14 - Fails statement 2- Even after Bernard hears Albert’s statement, he still doesn’t know which month it is in July 16 - passes August 14 - Fails statement 2- Even after Bernard hears Albert’s statement, he still doesn’t know which month it is in August 15 - Fails statement 3- Even after Albert hears Bernard’s statement, he still doesn’t know on which day in August the birthday falls August 17 - Fails statement 3- Even after Albert hears Bernard’s statement, he still doesn’t know on which day in August the birthday falls ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [April 14, 2015, 5:17pm UTC](https://boards.straightdope.com/t/birthday-logic-question/717622/60 "2015-04-14T17:17:18Z")
I hope it’s OK if I eliminate the spoiler tag at this late time… > [@Richard\_Pearse](#): > > 16th July. > > 1. Albert knows that Bernard does not know. > > This eliminates May and June as they are the two months with unique dates. > > 1. Bernard now knows the birthday. > > It can’t be the 14th, or Bernard would still not know. It must be the 15th, 16th, or 17th. > > 1. Albert now also knows the birthday. > > It must be the 16th of July. If Albert had been told August then he would still not know if it was the 15th or 17th, seeing as eliminating the 14th allows Albert to know, it must be July, therefore must be the 16th. I disagree with this - I don’t see why you say “This eliminates May and June as they are the two months with unique dates.” Here’s my analysis: Albert knows the month, Bernard knows the day. Albert knows that Bernard doesn’t know, that rules out the 18th and 19th days because those are unique, and if one of those were it, Bernard would have known right away. Bernard knows that Albert has ruled out the 18th and 19th, and we know that with his month information, Bernard now knows the exact date. The only way this can happen is if the month is June, and the only option left for June is the 17th. June 17. [Previous page](https://boards.straightdope.com/t/birthday-logic-question/717622.md?page=2) [Next page](https://boards.straightdope.com/t/birthday-logic-question/717622.md?page=4)