Competition problem — Combinatorics

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Combinatorics Difficulty 5.3 AIME, harder Find the answer

Alberto and Bernardo recently became friends with Carol, and they want to know when her birthday is. Carol gave them a list of 12 possible dates:

- January 4;
- March 8;
- June 7;
- October 7;
- January 5;
- April 8;
- June 5;
- October 4;
- January 11;
- April 9;
- July 13;
- October 8.

Then, Carol played a game by telling Alberto and Bernardo, separately, the month and the day of her birthday, respectively, and asked them to talk to each other to figure out her birthday.

It wouldn't be very fun if her birthday was in March or July, as there is only one possible date in each of these months among the options, and Alberto would immediately guess it. It would also be quick if her birthday was on January 11, April 9, or July 13, as these numbers only appear once in the list, and Bernardo could figure out the answer on his own. So, they started a conversation to try to find out.

Alberto says: I can't figure out Carol's birthday, but I can guarantee that Bernardo doesn't know either.

Bernardo says: At first, I didn't know the complete date of Carol's birthday, but now I do. Alberto: Then I also know when Carol's birthday is.

a) Why, in the first statement of Alberto, does he say that neither he nor Bernardo knows the answer yet?

b) Why did Bernardo say that at first he didn't know the answer, but after Alberto's statement, Bernardo figured out Carol's birthday?

c) When is Carol's birthday?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

a) Initially, Alberto had no way of knowing the answer, as except for March and July, all months have more than one possible date. So Carol's birthday is not in March or July. However, since he assured that Bernardo did not know, this means that the month he was informed of does not have an exclusive date in the set, that is, it cannot be January (11 is a date that only January has in the list), nor April (9 is a date that only April has in the list), nor July (for the 13th). And the options now are:

- 4th of January;
- 8th of March;
- 7th of June;
- 7th of October;
- 5th of January;
- 8th of April;
- 5th of June;
- 4th of October;
- 11th of January;
- 9th of April;
- 13th of July;
- 8th of October.

b) Now, when Bernardo says he didn't know but now knows the date, it is likely that the number he received appeared in more than one month in the list, and with the elimination of January, April, and July (from the previous item), the options are reduced to June and October. Moreover, it cannot be 7, as this would still leave some doubt. Thus, the list is:

c) Finally, Alberto also figures it out. Therefore, the date must be an option that is not common between June and October and unique in its respective month. Thus, since October has two other date options (4 and 8, which are only in October, but the doubt would still remain), Carol's birthday is on the 5th of June.
- 4th of January;
- 5th of January;
- 11th of January;
- 8th of March;
- 8th of April;
- 9th of April;
- 7th of June;
- 5th of June;
- 13th of July;
- 7th of October;
- 4th of October;
- 8th of October.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.