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Cheryl and the Perfect Logicians

In real life, making deductions from what other people say - or don't say - is not an exact science

Here is a variation on a very old puzzle. 

Cheryl has two white stickers and one black sticker. Her friends Albert and Bernard are sat facing each other, and without them seeing, Cheryl puts one of her stickers on Albert's forehead and one on Bernard's, keeping the third sticker hidden.  Albert and Bernard can both see the other man's sticker but not his own.  Cheryl tells her friends there will be a prize for the first who is able to announce with 100% confidence the colour of the sticker on his own forehead.  After a few moments of silence, Albert announces the colour of his sticker.  What is it?

The answer to this puzzle is at the end of the blog.  But I was reminded of it by the news that another much harder puzzle about Albert, Bernard and Cheryl has just gone viral.

Here it is if you missed it (I have reworded the original slightly for clarity):

Albert and Bernard want to know when Cheryl's birthday is.  She 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 whispers to Albert which month her birthday is in, and whispers to Bernard which day her birthday is on.

Albert says: I don't know when Cheryl's birthday is, but I can be certain that Bernard does not know either.

Bernard then says: At first I didn't know when Cheryl's birthday is, but now I do know.

At which point Albert says:  Then I also know when Cheryl's birthday is.

I'm not going to go into the solution - you will find it discussed widely on the web, in particular in Alex Bellos' excellent blog for The Guardian. 

I just want to make a couple of observations about this birthday puzzle, which was part of an Olympiad paper set for 14-15 year old Singapore pupils.

The first is that some of the media coverage has implied that Cheryl's birthday puzzle is the sort of thing that 15 year old Singaporeans can do and that we Europeans can't.  "Oh dear, no wonder we British are so hopeless at maths" seems to be the underlying message. This is nonsense of course.  Take a look at the Olympiad puzzles that our brighter teenagers are given to solve and you'll find many that are just as hard.  

Apparently the Singapore Olympiad is sat by 40% of their teenagers, but what we don't know is what proportion of those 40% got the Cheryl question right.  Since it was the penultimate question on the test - and questions at the end are always the hardest - I am going to guess that maybe only 10% of students got it right.  So that would  suggest only 4% of Singapore teenagers can answer this question.  That's still impressive, but it doesn't make them a nation of geniuses.

My other observation is that the Cheryl birthday poser is an example of a type of logic puzzle that only works if the participants are perfect logicians. A perfect logician is somebody who, when presented with some information, is able to accurately make every possible logical deduction from that information - Sherlock Holmes meets Spock, in other words.  Do such people really exist, or are they confined to puzzle-land?

A few years ago, I thought I'd try to find out, using the black and white sticker puzzle at the start of this blog. 

I used to run a management workshop on problem solving at The Civil Service College in Sunningdale. As part of the workshop,  I would ask two volunteers to sit opposite each other, and I would then (without them seeing the colour) put white stickers on each of their foreheads.  Remember, there are two white stickers and one black sticker. When asked if they could state their own sticker's colour, the two adults were always stumped.  Their reasoning was: "I don't know what my sticker is because the one I can see is white, which means that mine could be black or white".  They never made the next step, which was to put themselves in the mind of the other participant.  

And that is a relatively simple logic puzzle! It shows that in reality, perfect logicians are rare, and unless you are certain that the participants are perfect logicians, these Albert, Bernard and Cheryl puzzles are unsolvable.

But that's not to say great logicians aren't out there. On one occasion when running my workshop, there was a delegate who was blind.  As usual, the two volunteers facing each other were unable to deduce the colour of their own sticker.  Then something remarkable happened.  The blind delegate announced: "I know the colour of both of the stickers.  They are both white." Her reasoning was as follows: if either of the stickers were black, then one of the delegates would have revealed his colour by now.  The fact that neither of them had done so meant that they must both be white.

It was a smart bit of logic, and I wondered if for once being blind had been an advantage to her.  Undistracted by the ability to see any stickers from which to glean clues, she was forced to rely entirely on her ability to reason.


Answer to the opening puzzle: If Albert's sticker had been black, then Bernard would have known instantly that his was white.  The fact that Bernard hesitated meant Albert's must be white.  (Bernard could have made the same deduction given Albert's hesitation. What is not clear is how long a perfect logician needs to wait before being able to announce their deduction.)