>>13683875Albert's information: the month
Bernard's information: the day
May: 15, 16, 19
June: 17, 18
July: 14, 16
August, 14, 15, 17
14: July, August
15: August, May
16: July, May
17: August, June
18: June
19: May
Albert is not able to deduce Cheryl's birthday from his information, the month.
This excludes any months with only a single day (there are none).
Bernard is, initially as Albert says, unable to find Cheryl's birthday from his information, the day. This excludes days only associated with a single month, June 18 and May 19:
May: 15, 16
June: 17
July: 14, 16
August, 14, 15, 17
14: July, August
15: August, May
16: July, May
17: August, June
Now the June set has only one possibility, June 17. If the month was June Bernard would know Cheryl's birthday is June 17, but he doesn't yet. So exclude June 17.
May: 15, 16
July: 14, 16
August, 14, 15, 17
14: July, August
15: August, May
16: July, May
17: August
August set has one possibility...if the day was 17 Bernard would know August 17 is Cheryl's birthday. He doesn't, as Albert says, so remove this set from the possibilities.
May: 15, 16
July: 14, 16
August, 14, 15
14: July, August
15: August, May
16: July, May
Earlier we excluded May 19 because day 19 was exclusive to May.
If the day was 19, Bernard would have known the month is May.
If the month was May, Albert would known the day is 19.
The contrapositives of these statements:
If the month is not May, then the day is not 19
If the day is not 19, then the month is not May
The day was not 19 because if it was then Bernard would have been able to deduce the month as May earlier before May 19 was excluded.
So we exclude May:
July: 14, 16
August, 14, 15
14: July, August
15: August
16: July
From similar contrapositives of excluding August 17, If the day is not 17, then the month is not August:
16: July
Thus Cheryl's birthday is July 16
