(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.
- 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.
- 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.