17 APR 2015
Ms. Cheryl's Birthday Problem
A Solution

Cheryl's Birthday Problem
To sort this out we take each comment in turn and analyze it.

[1st Comment Question] How is it that Albert can possibly know that Bernard is ignorant of Sheryl's Birthday?

[Answer] The key is to notice that two of the month-day pairs are so that the day value of those pairs only occur on a single month. Writing the problem in table form below shows MAY-19 and JUN-18 fit this pattern.

From Benard's perspective either of those two dates would be a dead give away because the number 18 and the number 19 only belong to a single month.

Albert does not have this sort of give away because there are NO months that occur only once; this explains why he said he does not know the answer.

But how does this tell us why Albert can be sure that Bernard does NOT KNOW answer?

The reason is: Cheryl must have told Albert that the month of her birthday was July or August and that completely ruled out Benard's give away solutions.
      14  15 16 17 18 19  TOT
  MAY     XX XX       XX  03 
  JUN           XX XX     02 
  JUL XX     XX           02 
  AUG XX  XX    XX        03 
  TOT 02  02 02 02 01 01     

[2nd Comment Question] How in the world can Bernard then be sure he knows the Cheryl's birthday after just listening to Albert?

[2nd Comment Answer] From listening to Albert Benard knows that May and June are out the window. And the added benefit of this is when you just consider the months July and August you get a much smaller table as illustrated below.

When we analyze the smaller table like we did the one above we see why Bernard now knows for sure the birthday.

The reason is: The days 15 or 16 or 17 only appear in a single month. The day 14 appears in two months. So Benard really has three chances to know Cheryl's birthday correctly.
      14  15 16 17 TOT
  JUL XX     XX    02 
  AUG XX  XX    XX 03 
  TOT 02  01 01 01      

[3rd Comment Question] And finally, how does Albert get to say, "Me Too!"?

[3rd Comment Answer] The reason is the same as before Albert understands what allowd Bernard to be sure of Cheryl's birthday. So he too can reduce the size of the table even more as illustrated below and Cheryl's birthday is now a give away for him.
      15 16 17 TOT
  JUL    XX    01 
  AUG XX    XX 03 
  TOT 01 01 01      

[4th Comment Question] So is this enough for you also to know Cheryl's birthday? [4th Comment Answer] Yes! (I hope.)

[A Final Comment] I did an analysis of this problem using the same logic as above but did NOT USE THE month/day tables and I failed to realize that there were TWO POSSIBLE unique day choices for the 1st comment. That error of only thinking there was one combined with the error of not doing the corresponding analysis of the smaller table lead me down a path that resulted in my choosing the wrong birthday.

I did my analysis on purpose before watching a video so I would not be biased by an actual solution. My error jumped right out at me after watching one of my favorite math folks YouTube.COM videos (Simon Pampena) of this puzzle. And this post is my statement of the logic flow that resulted. And of course my solution now looks just like how Simon did it.

This page last updated: 17 April 2015
This page created: 17 April 2015