Gregory Pettigrew (etherial) wrote,
Gregory Pettigrew
etherial

Three Logicians walk into a bar...

There's a logic puzzle that's been going around that has a bunch of my friends stumped. I don't know if it's really from a primary school assignment in Singapore (the syntax certainly reflects Singapore's mixed heritage), but I solved more complicated logic puzzles all the time in elementary school, so maybe.

Albert and Bernard just become friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates.

May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, August 17

Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively.

Albert: I don't know when Cheryl's birthday is, but I know that Bernard does not know too.

Bernard: At first I don't know when Cheryl's birthday is, but I know now.

Albert: Then I also know when Cheryl's birthday is.

So when is Cheryl's birthday?



We have a bunch of initial facts and then 3 statements.

0.

I'm going to take it as given that Albert knows the month and Bernard knows the day. Other people do not assume this and arrive at the same answer, but the puzzle uses the word "respectively", so I conclude the first person (Albert) gets told the first thing (month) and the second person (Bernard) gets told the second thing (day).

Albert knows the month and knows the day is one of {14, 15, 16, 17, 18, 19}
Bernard knows the day and knows the month is one of {May, June, July, August}
We know the date is one of {May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, August 17}

1. Albert: I don't know when Cheryl's birthday is, but I know that Bernard does not know too.

Albert knows that Bernard doesn't know. The only way Albert can be certain of this is if the correct month does not have either of the unique days (18 and 19), so we can eliminate May and June.

Albert knows the month and knows the day is one of {14, 15, 16, 17, 18, 19}
Bernard knows the day and knows the month is one of {May, June, July, August}
We know the date is one of {May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, August 17}

2. Bernard: At first I don't know when Cheryl's birthday is, but I know now.

From the days that are left, it now must be a unique day, so we can eliminate 14.

Albert knows the month and knows the day is one of {14, 15, 16, 17, 18, 19}
Bernard knows the date.
We know the date is one of {May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, August 17}

3. Albert: Then I also know when Cheryl's birthday is.

From the months that are left, it now must be a unique month, so we can eliminate August.

Albert knows the date.
Bernard knows the date.
We know the date is July 16.

One thing that confuses people is what is going on in Step 1. Albert is passing information to Bernard. Of course Bernard doesn't know: the first statement is not Bernard saying "I know what Cheryl's birthday is". They know they're in a logic puzzle and need to collaborate in order to get enough information on the table for them (and you the reader) to solve it. Let's look at it again now that we know their starting information:

0.

Albert knows the date is either July 14 or July 16.
Bernard knows the date is either May 16 or July 16.
We know the date is July 16.

1. Albert knows Bernard is staring at either July/August 14 or May/July 16. Albert knows that Bernard cannot eliminate either date without him and allows Bernard to eliminate May 16. If Bernard has July/August 14, this doesn't actually help Bernard, but either way it gives Bernard the ability to help Albert.

Albert knows the date is either July 14 or July 16.
Bernard knows the date.
We know the date is July 16.

2. Bernard allows Albert to eliminate the 14th.

Albert knows the date.
Bernard knows the date.
We know the date is July 16.
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