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How to solve it: Cheryl's birthday puzzle part two, Denise's revenge How to solve it! Cheryl's birthday puzzle part two: Denise's revenge
(about 1 hour later)
Guzzlers, how did you get on?Guzzlers, how did you get on?
Let me first restate the problem. Albert, Bernard and Cheryl became friends with Denise, and they wanted to know when her birthday is. Denise gave them a list of 20 possible dates.Let me first restate the problem. Albert, Bernard and Cheryl became friends with Denise, and they wanted to know when her birthday is. Denise gave them a list of 20 possible dates.
Denise then told Albert, Bernard and Cheryl separately the month, the day and the year of her birthday respectively.Denise then told Albert, Bernard and Cheryl separately the month, the day and the year of her birthday respectively.
Albert: I don’t know when Denise’s birthday is, but I know that Bernard does not know.Albert: I don’t know when Denise’s birthday is, but I know that Bernard does not know.
Bernard: I still don’t know when Denise’s birthday is, but I know that Cheryl still does not know.Bernard: I still don’t know when Denise’s birthday is, but I know that Cheryl still does not know.
Cheryl: I still don’t know when Denise’s birthday is, but I know that Albert still does not know.Cheryl: I still don’t know when Denise’s birthday is, but I know that Albert still does not know.
Albert: Now I know when Denise’s birthday is.Albert: Now I know when Denise’s birthday is.
Bernard: Now I know too.Bernard: Now I know too.
Cheryl: Me too.Cheryl: Me too.
So when is Denise’s birthday?So when is Denise’s birthday?
The solution:The solution:
You solve this in the same way as you do the Cheryl problem. Each sentence gives you a clue as to what to eliminate. Yet the Denise problem is more involved, and pushes your brain to think in different directions.You solve this in the same way as you do the Cheryl problem. Each sentence gives you a clue as to what to eliminate. Yet the Denise problem is more involved, and pushes your brain to think in different directions.
Let’s take it line by line:Let’s take it line by line:
Albert: I don’t know when Denise’s birthday is, but I know that Bernard does not know.Albert: I don’t know when Denise’s birthday is, but I know that Bernard does not know.
Albert has the month so of course he doesn’t know. But if he knows that Bernard doesn’t know, then he knows that Bernard does not have either 11 or 12, since these are the only numbers that appear only once in the selected birthdays. If Bernard had 11 or 12 he would know, and the date would either be 11 April 2003 or 12 June 2002. We can deduce therefore that Albert does not have June or April.Albert has the month so of course he doesn’t know. But if he knows that Bernard doesn’t know, then he knows that Bernard does not have either 11 or 12, since these are the only numbers that appear only once in the selected birthdays. If Bernard had 11 or 12 he would know, and the date would either be 11 April 2003 or 12 June 2002. We can deduce therefore that Albert does not have June or April.
Let’s eliminate June and April: this deletes five dates.Let’s eliminate June and April: this deletes five dates.
Bernard: I still don’t know when Denise’s birthday is, but I know that Cheryl still does not know.Bernard: I still don’t know when Denise’s birthday is, but I know that Cheryl still does not know.
Bernard has a number. If that number appears only once in the remaining dates then we can eliminate it, since a unique number would mean he knows the answer, but he doesn’t. The numbers 15 and 17 appear only once, so we can strike them out: that’s 15 May 2001 and 17 Feb 2001.Bernard has a number. If that number appears only once in the remaining dates then we can eliminate it, since a unique number would mean he knows the answer, but he doesn’t. The numbers 15 and 17 appear only once, so we can strike them out: that’s 15 May 2001 and 17 Feb 2001.
But Bernard also knows that Cheryl doesn’t know. The only way Cheryl could know is if she has 2001, since there is only one 2001 option left. So Bernard does not have 13, and we can eliminate its two remaining appearances: 13 March 2001 and 13 Jan 2003.But Bernard also knows that Cheryl doesn’t know. The only way Cheryl could know is if she has 2001, since there is only one 2001 option left. So Bernard does not have 13, and we can eliminate its two remaining appearances: 13 March 2001 and 13 Jan 2003.
Cheryl: I still don’t know when Denise’s birthday is, but I know that Albert still does not know.Cheryl: I still don’t know when Denise’s birthday is, but I know that Albert still does not know.
Cheryl not knowing when the birthday is does not provide us with new information, but if she knows Albert still doesn’t know, then Albert cannot have a month that appears only once in the remaining options. The only month that appears once is Jan, which means that Cheryl does not have 2004.Cheryl not knowing when the birthday is does not provide us with new information, but if she knows Albert still doesn’t know, then Albert cannot have a month that appears only once in the remaining options. The only month that appears once is Jan, which means that Cheryl does not have 2004.
Albert: Now I know when Denise’s birthday is.Albert: Now I know when Denise’s birthday is.
Albert must have a month that appears only once in the remaining options. So we can delete the two March dates.Albert must have a month that appears only once in the remaining options. So we can delete the two March dates.
Bernard: Now I know too.Bernard: Now I know too.
Bernard must have a number that appears only once. We can delete the 16s.Bernard must have a number that appears only once. We can delete the 16s.
Cheryl: Me too.Cheryl: Me too.
There is only one date left: 14 May 2002.There is only one date left: 14 May 2002.
The Easter eggThe Easter egg
Joseph Yeo smuggled in a mathematical nugget, perceptible only to the truly geeky.Joseph Yeo smuggled in a mathematical nugget, perceptible only to the truly geeky.
Did you recognise anything special about 14/5/2002?Did you recognise anything special about 14/5/2002?
These numbers are a familiar bedfellows: there are 2002 ways to pick 5 objects from a group of 14.These numbers are a familiar bedfellows: there are 2002 ways to pick 5 objects from a group of 14.
In the symbology of combinatorics: 14C5 = 2002.In the symbology of combinatorics: 14C5 = 2002.
The year previous to 2002 that is expressible in the form nCk where n and k are whole numbers is 1953 = 63C2, and the next one is 2016 = 64C2 followed by 2024 = 24C3. (Excluding trivial cases when k = 1).The year previous to 2002 that is expressible in the form nCk where n and k are whole numbers is 1953 = 63C2, and the next one is 2016 = 64C2 followed by 2024 = 24C3. (Excluding trivial cases when k = 1).
I hope you enjoyed it. And see you back in two weeks.I hope you enjoyed it. And see you back in two weeks.