This logical deduction puzzle, inspired by the famous “Cheryl’s Birthday,” revolves around narrowing down possibilities from a short conversation between two people with partial information.

The Setup

Your boss A’s birthday is one of the following 10 dates:

  • March: 4, 5, 8
  • June: 4, 7
  • September: 1, 5
  • December: 1, 2, 8

You are told only the month, while your colleague C is told only the day.

Everyone knows the list of 10 possible dates.

Step 1: You Say

“I don’t know A’s birthday, and I know C doesn’t know it either.”

This tells us your month cannot be one with any uniquely identifying days.

Let’s look at the days across months:

  • 1 → September, December
  • 2 → December
  • 4 → March, June
  • 5 → March, September
  • 7 → June
  • 8 → March, December

Some days are unique:

  • 7 only appears in June
  • 2 only appears in December

If your month were June (with day 7) or December (with day 2), then C might know the birthday right away.

So when you confidently say, “I know C doesn’t know,” it rules out June and December as your month.

You’re left with March and September.

Step 2: C Says

“At first I didn’t know A’s birthday, but now I do.”

C must have heard a day that was not unique at first, but now can be uniquely associated with one date, given the month must be March or September.

Let’s filter the list down to March and September dates:

  • March 4, 5, 8
  • September 1, 5

So the possible days are: 1, 4, 5, 8

C must have one of these.

Now let’s test which days can identify a unique date from this reduced list:

  • 1 → Only in September
  • 4 → Only in March
  • 5 → Appears in both March and September → not unique
  • 8 → Only in March

So if C had 5, they’d still be confused. But if C had 1, 4, or 8, they can now know the date.

Therefore, C must have seen 1, 4, or 8.

Step 3: You Say

“Now I know it, too.”

You must have been told March or September, and now that C figured it out, you must also be able to.

If your month were March, remaining candidates: March 4, 5, 8
Only 4 and 8 are uniquely identifying after step 2.

If your month were September, remaining: September 1, 5
Only 1 is uniquely identifying.

So, to be sure, your month must be one where only one date remains valid after step 2.

  • If your month was March, and the day was 4 or 8 → still ambiguous
  • If your month was September, and the day was 1 → unique

Hence, A’s birthday is September 1.

Final Answer

September 1

Reference