# Arnie and Barney: When is Barney's birthday?

The following twelve statements were made on a certain day X during the last 100 years. At least seven of these statements were true on day X:

1. Exactly one month from now it will be Thanksgiving.
2. Arnie was born in San Francisco.
3. Barney is in his twenties.
4. Exactly one month from now is Independence Day (July 4).
5. Exactly one month from now is Easter.
6. Barney's favorite movie is Akira Kurosawa's "Rashomon".
7. Barney was born in New York City.
8. Arnie celebrated his 30th birthday yesterday.
9. Barney was born in Seattle.
10. Arnie was born in Miama.
11. Barney is older than Arnie.
12. Exactly one month from now is Christmas.

On which day (day+month+year) was Barney born?

(1) First we derive that at most three out of the eight statements 1, 2, 4, 5, 7, 9, 10, 12 can be true.

• Statements 1, 4, 5, 12 are pairwise contradictory. Thanksgiving is in late November, Independence Day in July, Easter in spring, Christmas in December. Hence at most one out of 1, 4, 5, 12 is true.

• Statements 2 and 10 on Arnie's birthplace are contradictory. Hence at most one of 2 and 10 is true.

• Statements 7 and 9 on Barney's birthplace are contradictory. Hence at most one of 7 and 9 is true.

(2) Since altogether there are at least seven true statements, we conclude that the remaining four statement 3, 6, 8 and 11 all must be true. To summarize:

• 3, 6, 8 and 11 are true
• exactly one of 1, 4, 5, 12 is true
• exactly one of 2, 10 is true
• exactly one of 7, 9 is true.

(3) Statements 3, 8, 11 state that Arnie's mathematical age (counted in years) is larger than Barney's mathematical age, although Barney is older than Arnie.

• Hence Barney was born in an eastern time zone (very early in the day, let us say just after midnight).
• Arney was born in a time zone farther to the west, later than Barney, but still on the day before Barney.
• This implies that statements 2 and 7 are true: Barney was born in New York City and Arnie was born in San Francisco.

(4) By the true statement 8, Arnie's 30th birthday was yesterday. By the true statement 3, Barney's 30th birthday will start in the future. This means that the birthdays were consecutive days in the year when they both were born, but not in the year with day X. The only possible reason for this discrepancy must be a February 29 in the year with day X.

• Hence Barney was born on March 1 in New York City (say at 0:01am).
• Arnie was born later (say at 11:55pm) on February 28 in San Francisco.
• Day X is a February 29.

(5) Exactly one of statements 1, 4, 5, 12 must be true, and the only remaining possibility is statement 5 with Easter. Since statement 6 on Rashomon is true and since Rashomon came out in 1950, the year with day X is 1950 or later. We summarize:

• In the year with day X, Easter is on March 29
• The year with day X is 1950 or later.
• The year with day X is a leap year

(6) Looking up the Easter days in http://www.maa.mhn.de/StarDate/publ_holidays.html, we see that

• Day X is 29 February 1964
• Arnie was born 28 February 1934
• Barney was born on 1 March 1934.
• Good work. Your point #3 I think, was the main key for all the pieces falling together. Feb 4, 2015 at 16:40

My first variant...

Barney was born on 29 February, 1844 (Year may be different)

Today is 1st March 1934

1) Arnie was born in San Francisco / Arnie was born in Miama.

2) Barney was born in New York City / Barney was born in Seattle.

3) Barney's favorite movie is Akira Kurosawa's "Rashomon".

4) Exactly one month from now is Easter. (1st April 1934)

5) Barney is in his twenties (yep, he celebrates his birthday once in 4 years)

6) Arnie celebrated his 30th birthday yesterday. (28 February 1934, he celebrates his birthday every year)

7) Barney is older than Arnie. (yep, for many-many years)

My second variant... ("Rashomon" won't stop me xD)

Barney was born on 1 March 1934.

Today: 29th February 1964.

1) Arnie was born on 28 February 1934 in Miama at 23:30 (4:30 GMT)

2) Barney was born on 1 March 1934 in Seattle at 00:30 (3:30 GMT)

3) Barney's favorite movie is Akira Kurosawa's "Rashomon".

4) Exactly one month from now is Easter. (29 March 1964)

5) Barney is in his twenties (He will hit 30 tomorrow)

6) Arnie celebrated his 30th birthday yesterday. (28 February 1964)

7) Barney is older than Arnie. (technically, yes)

• This does not quite work: "Rashomon" came out in 1950 (en.wikipedia.org/wiki/Rashomon), and cannot be Barney's favorite movie in 1934. ------ Rashomon was clearly put into the puzzle to restrict the possibilities for day X. Feb 3, 2015 at 14:18
• By what logic did you determine your #4 to be true? This would be pushing the definition of age would it not? Even if you are born on a leap day you still age once a year. Feb 4, 2015 at 7:16
• @Octopus I added the second answer. Seems more realistic. Rashomon and right age definition are here. Feb 4, 2015 at 8:22
• It's not clear to me how you arrived at these conclusions. How have you ruled out that Arnie was born in S.F., for example? Feb 4, 2015 at 8:48
• @Gerhard, how do you know for sure that statement #6 is necessarily true? I believe it probably was put into the puzzle to restrict the possibilities, but maybe its a red herring. Feb 4, 2015 at 8:52

I don't have a complete answer yet, but so far I have been able to determine that:

2 & 10 are contradictory so at least 1 of them is false

7 & 9 are contradictory so at least 1 of them is false

3, 8 & 11 are contradictory. at least 1 of them is false.

1, 4, 5 & 12 are contradictory. at least 3 of them are false

even though the date for thanksgiving has changed throughout history none of the dates are close enough to have fallen on the same date (afaik).

that means we have at least 6 false statements, which contradicts the original question that stated there are at least 7 true statements.

therefore some part of my understanding is likely mistaken (?)

• 3, 8 & 11 are questionable. Those who were born on 29 February can celebrate their birthdays once in a 4 year, that allows them to be 10 years old, while being 40 years old for real. The next fact is about places - they have exactly 2 hours of difference (because of time zones), that allows someone to be born earlier than someone else, but on the next day. Feb 4, 2015 at 7:50
• @Xellos, being born on a leap day doesnt stop you from aging once a year. that may be part of the solution, but really that's just being silly. maybe if the op said he's only celebrated 20 birthdays, that would be different, but I am going to assume that isn't the right way to solve it. Feb 4, 2015 at 7:52