During the course of the year 2000, two brothers Andrew and Brady set out to research the family tree and began by listing the dates of birth of all the family members born during the period covering January 1, 1900 to December 31, 1999 inclusively.

Each date was recorded as three pairs of digits, written in succession to form a positive six digit decimal integer with a zero preceding any single digit day, month or year. Andrew followed the dd-mm-yy format, while Brady followed the mm-dd-yy format. For example, May 7, 1998 would be written as 070598 and 050798 respectively by Andrew and Brady.

Comparing the two different integers representing Chloe’s birth date, they noticed that one of them was a perfect square, and the absolute difference between them was a nonzero perfect square. The same was true for their versions of the birth date of Chloe’s sister Denise, who is younger by between one and two years.

When was Chloe born? What was the birth date of Denise?

Note: Try to derive a non computer-assisted solution, although computer program/spreadsheet solutions are welcome.

Using an Excel spreadsheet I found four dates that satisfy the conditions for Chloe's birth date:
Feb 13, 1916
Jul 18, 1924
Nov 22, 1925
Jan 12, 1936

Only the middle two satisfy the additional relationship between Chloe's and Denise's birth dates.  So Chloe's birthdate is Jul 18, 1924 and Denise's is Nov 22, 1925.

Interestingly, the absolute value of the differences between the two integers for a given date are the same regardless of the year and the only non-zero difference that is a perfect square is 108,900 which is yielded by only 13 dates per year: Jan 12, Feb 13, Mar 14, Apr 15, May 16, Jun 17, Jul 18, Aug 19, Sep 20, Oct 21, Nov 22, Dec 1, and Dec 23.  This narrows down the possibilities considerably.

I was in the middle of posting this when Charlie posted his solution.  After seeing his results which included Dec 1, 1936 which I did not originally have, I went back to my spreadsheet and, sure enough, it was there.  I just missed it the first time around.

Edited on December 23, 2008, 1:53 pm

Posted by Sing4TheDay on 2008-12-23 13:47:39