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Jack tells Art and Ben: my birthday is on one of these 10 days:
Mar 4 ; Mar 5 ; Mar 8
Jun 4 ; Jun 7
Sep 1 ; Sep 5
Dec 1 ; Dec 2 ; Dec 8

Jack then tells Art the day, and Ben the month.
Then he asks them: "Do you know when my birthday is?"

Art says: "I don't know"
Ben says: "I knew that you didn't know"
Art says: "Now I know"
Ben says: "Ah, now I also know"

Based on the above dialogue and dates, can you figure out Jack's birthday?

Sep 1

Correct.

In case someone asks me later and I forgot(!), this is the solution:

Dialogue can be reduced to:
Ben says: "I know that Art doesn't know"
Art says: "Now I know"
Ben says: "Ah, now I also know"

(notice that there is only one "2" and one "7")

Ben says: "I know that Art doesn't know"
Eliminates Jun and Dec, else Ben wouldn't be sure,
since Art could have Jun 7 or Dec 2.

Art says: "Now I know"
Knowing now that month is Jan or Sep, Jan 5 and Sep 5 are eliminated,
since being in both months, Art would not know if he was given 5.

Ben says: "Ah, now I also know"
Only way Ben can know is if he was given Sep, since only "1" is possible;
if given Jan, he'd have the 4th and 8th as possibilities.

So it's Sep 1.

Glad you did that because I was going to ask, instead of scratching my head too much.

After all that I'm still scratching.