Cheryl's Birthday and Iterated Knowledge

The problem. Cheryl gives Albert the month of her birthday and Bernard the day, keeping the pair secret from each. The ten possible dates are: May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, August 17. Albert says: "I don't know when Cheryl's birthday is, but I know Bernard doesn't know either." Bernard then says: "At first I didn't know, but now I do." Albert then says: "Now I know too." What is Cheryl's birthday?

This puzzle is a favorite in analytical interviews not for the arithmetic (there is none) but for testing whether a candidate can track what a statement reveals about the speaker's own information, three layers deep.

Step 1: Albert's first statement rules out unique days

Albert knows the month but not the day, and he claims to know for certain that Bernard (who has only the day) can't determine the date either. Bernard would immediately know the date if his day appeared only once in the list — days appearing once are 19 (only in May 19) and 18 (only in June 18). For Albert to be certain Bernard doesn't know, Albert's month must not contain either of those unique days — because if it did, there'd be a chance the day was 19 or 18 and Bernard would already know. May and June both contain a uniquely-occurring day (19 and 18 respectively), so Albert's month can't be May or June. Albert's statement tells us his month is July or August, eliminating May 15, May 16, May 19, June 17, June 18 from the list. Remaining: July 14, July 16, August 14, August 15, August 17.

Step 2: Bernard's statement rules out ambiguous days

Bernard now knows Albert's constraint (month is July or August) and, combined with his own day, can now determine the date for certain. Look at the remaining list: day 14 appears twice (July 14, August 14) — if Bernard's day were 14, he still couldn't distinguish between them, so he wouldn't be able to say "now I know." Days 16, 15, 17 each appear only once in the remaining list. So Bernard's day must not be 14, eliminating July 14 and August 14. Remaining: July 16, August 15, August 17.

Step 3: Albert's second statement breaks the last tie

Albert now also knows the date for certain, given his own month. Look at the remaining list by month: July has only July 16 left — if Albert's month were July, he'd already know the answer is July 16 (unique). August has two remaining candidates, August 15 and August 17 — if Albert's month were August, he still couldn't distinguish between them. Since Albert says he now knows, his month must be July, because only July narrows to a single remaining date.

$$\textbf{Cheryl's birthday is July 16.}$$

The general technique: statements as filters on the search space

Each statement in this puzzle isn't new factual information about the date directly — it's a statement about what the speaker does or doesn't know, which indirectly rules out candidates by revealing which entries in the list would have made the speaker's knowledge (or ignorance) impossible. The systematic way to solve any puzzle in this family: maintain the full candidate list, and after each statement, cross out every candidate that is inconsistent with the speaker being able to truthfully make that exact statement — not candidates that contradict the date directly, but candidates that would have given the speaker different knowledge than they claimed to have.