Treebeard's Stumper Answer 5 October 2001

Palindromic Dates

Thanks Donna, you pointed out that last Tuesday, October 2, 2001, was a palindromic date. Written in the usual American way (mm-dd-yyyy), that's 10-02-2001. The digits are the same forward and backward! The obvious stumper is when will this happen again, and when did it happen last? In Europe, they write dates as dd-mm-yyyy, so Tuesday was 02-10-2001. The International Standard is yyyy-mm-dd, so it was 2001-10-02. Which way of writing dates will produce the next palindromic date? What if we all decide to write yy instead of yyyy?

The next palindromic date after October 2, 2001 (10-02-2001) will be January 2, 2010 or 01-02-2010 (as mm-dd-yyyy). JP was the first Dunn Middle School student to figure that this last happened way back on August 31, 1380 or 08-31-1380. It's different if we write our dates as dd-mm-yyyy or yyyy-mm-dd, though both give the same result. (Why? D'oh!) The next reversible date will then be February 20, 2002 or 20-02-2002. There are many more reversible dates if we write the year as yy instead of yyyy. For example, yesterday was October 11, 2001 or 10-11-01.

Notes:

Working back to the last palindromic mm-dd-yyyy date was the hard one. The constraint is that months can only be {1 to 12} and days can only be {1 to 28-31} depending on the month. So start working backwards.

mm-dd	yyyy
10-02	2001	this palindromic date
00-xx	2000	illegal month
xx-91	19xx	illegal day, same for 18xx, 17xx, 16xx, 15xx, 14xx
xx-31	13xx	possible day, so refine it
01-31	1310	possible, same for 1320, 1330, 1340, 1350, 1360
07-31	1370	possible, and better
08-31	1380	possible, and the best so far
09-31	1390	looks good, but September only has 30 days so it's illegal
10-31	1301	wups, it got lower, all done

Similar trial and error found the other answers.

Graybear sent this complete answer that considers some other conventions:

I noticed it on the first of the month. One of my students asked the date so I wrote it on the board as 10-1-01 and noticed that it was palindromic and made up of only 1s and 0s. By writing the date in this format (mm-d-yy or mm-dd-yy), the first nine days of this month, as well as the 11th and the 22 are palindromic; then we'll have to wait until 2-01-02 (except if you write the day as 01, you'ld probably write the month as 02 so, for the purists, the next date would be 01-11-10).

If you write the date as mm-dd-yyyy, the most recent date was 10-02-2001 the next date is 01-02-2010. As dd-mm-yyyy, the last date was 10-02-2001 (last February 10th), the next date is 20-02-2002 (next February 20th).

Using the yyyy-mm-dd format gives the same palindromes as mm-dd-yyyy! (Why? Duh!)

As for yy formats, I mentioned mm-dd-yy above, for dd-mm-yy, the most recent was in September, 10-9-01, the next in November, 10-11-01.

And, last but not least, is yy-mm-dd. It works the same as dd-mm-yy. This is the format I use for files that I update and want to know what "version" each one is without opening it. For example, if I were to file a draft of this letter, the filename would be: 011008LtrToTreebeard. By doing this, the computer automatically sorts them "alphabetically" by date.

Good answer, Graybear. I use my own modified version of the ISO date standard to name my 1000s of digital photos and still stay with a classic MSDOS 8.3 character file name for my primative database. The 10th picture I took today on October 14, 2001 will become:

        1A14-010.jpg
                 ---    format  (jpg for photos)
             ---        picture (001-999, more than enough for a day)
          --            day     (01-31)
         -              month   (1-9, A-C in hex fashion: A is Oct, B is Nov, and C is Dec)
        -               year    (0-9, for 2000-2009)

I'll deal with 2010 when it happens! File naming conventions like this are essential, even if they create future personal Y2K-like problems. The International Standard (ISO 8601) is the right way to go if you are just starting a database with long filenames.

I thought about writing a computer program to find all palindromic dates by testing all possible dates, and then I realized I didn't have to. There are just 366 legal days in a year (including the occasional leap day), and each one translates into one unique palindromic year. Leap days are interesting, but it turns out that leap day palindromes are legal for both mm-dd-yyyy and dd-mm-yyyy schemes according to the rule that (non-century) years are leap years if the last two digits are divisible by four:

mm-dd-yyyy	02-29-9220	20/4=5,   it's legal
dd-mm-yyyy	29-02-2092	92/4=23, it's legal

There are also a few days missing from history. Because of the transition from the Julian to the Gregorian calendar, October 4, 1582 was followed by October 15 1582 in Catholic countries, and September 2, 1752 was followed by September 14, 1752 in Great Britain and the US. But these "missing days" are not potential palindomes.

I did write a small BASIC program to generate all palindromic dates and count the days between successive dates. Both ways of writing dates have 366 palindromic dates in the years between 0001 - 9999, but the distributions are quite different and have interesting patterns. The complete list is available on a separate Web page. Here's part of that list:

Count	Date	mm-dd-yyyy	Days since last	Date	dd-mm-yyyy	Days since last
77	Jul 31, 1370	07-31-1370	7366	24 Nov 1142	24-11-1142	375
78	Aug 31, 1380	08-31-1380	3684	05 Nov 1150	05-11-1150	2903
79	Oct 02, 2001	10-02-2001	226839	15 Nov 1151	15-11-1151	375
80	Jan 02, 2010	01-02-2010	3014	25 Nov 1152	25-11-1152	376
81	Nov 02, 2011	11-02-2011	669	06 Nov 1160	06-11-1160	2903
82	Feb 02, 2020	02-02-2020	3014	16 Nov 1161	16-11-1161	375
83	Dec 02, 2021	12-02-2021	669	26 Nov 1162	26-11-1162	375
84	Mar 02, 2030	03-02-2030	3012	07 Nov 1170	07-11-1170	2903
85	Apr 02, 2040	04-02-2040	3684	17 Nov 1171	17-11-1171	375
86	May 02, 2050	05-02-2050	3682	27 Nov 1172	27-11-1172	376
87	Jun 02, 2060	06-02-2060	3684	08 Nov 1180	08-11-1180	2903
88	Jul 02, 2070	07-02-2070	3682	18 Nov 1181	18-11-1181	375
89	Aug 02, 2080	08-02-2080	3684	28 Nov 1182	28-11-1182	375
90	Sep 02, 2090	09-02-2090	3683	09 Nov 1190	09-11-1190	2903
91	Oct 12, 2101	10-12-2101	4057	19 Nov 1191	19-11-1191	375
92	Jan 12, 2110	01-12-2110	3014	29 Nov 1192	29-11-1192	376
93	Nov 12, 2111	11-12-2111	669	10 Feb 2001	10-02-2001	295182
94	Feb 12, 2120	02-12-2120	3014	20 Feb 2002	20-02-2002	375
95	Dec 12, 2121	12-12-2121	669	01 Feb 2010	01-02-2010	2903
96	Mar 12, 2130	03-12-2130	3012	11 Feb 2011	11-02-2011	375

I also counted palindromic dates per century. There are nice patterns, and some interesting exceptions that I don't have time to explore now. There were 226,839 days between August 31, 1380 (08-31-1380) and January 2, 2010 (01-02-2010). The largest gaps are 259,698 days (mm-dd-yyyy) and 332,017 (dd-mm-yyyy). Stumpers remain.

Here are some links for further research:

• There's a French website about this very stumper, but it's posed in the European way (dd-mm-yyyy) for "Le 10 février 2001" (February 10, 2001 or 10-02-2001). It also has a table of palindromic years, but I'm sure they left out a few, eg 04-10-0140. Maybe the author has different assumptions?

• I kept all the leading zeros in my version of the stumper, but most of us drop them for months and days at least. Our shortcuts generate several more variant stumpers which are discussed at Ask Dr. Math and here.

• There's a reference site on the International Standard (ISO 8601). The Calendar FAQ and Calendar Zone have lots more calendar info.

• I wrote a small MSDOS BASIC program to figure all palindromic dates from 0001-9999. PALDAY.BAS is available for download (with BASIC source code) at Treebeard's Basic Vault.

• I've got more tricky calendar stumpers, including Five Fridays (12 Nov 99), Friday the 13th (13 Nov 1998), Full Moon Birthday (2 Oct 1998), and When is Washington's Birthday? (28 Feb 1997). Here's another: we all know about the sixties and the nineties, but what should we call this decade? The oh-ohs sounds about right.

Back to Stumper

Last modified .