Lots of solutions to this problem were submitted this month.

Clement from River Valley High School and Andrei of School 205, Bucharest, sent in the most complete solutions, with Andrei explaining his method clearly and Clement looking at the time between the dates. Clement's findings were very interesting it would be good to try to explain why the patterns of time between palindromic dates is as Clement describes. When you identify patterns in investigations and problems like this it is very useful to identify why the pattern works in terms of the mathematical problem or context itself.

Below is a solution based on Clement's and Andrei's work.

I used the listing method to solve this question.
My results are as below:

I observed that the palindromic date in February written in the British way is:
dd/mm/yyyy

where mm and yyyy are 02 and 2002 respectively:
dd/02/2002

It is read from left to right:
dd022002

This date must be read in the same manner from right to left:
200220dd

So, in February dd is 20 and the date is:
20/02/2002

I searched only for dates from this century:
dd/mm/20yy

and the date read from right to left:
dd/02/yyyy

So, this century all palindromic dates written in the British way are in February. Now, in search for other palindromic dates in this decade:
30/02/2003

But, this date does not exist and from now on the days will be 40, 50 etc. and these dates do NOT exist. So, I found the following date in 2010 :
01/02/2010

Now, I wrote the solutions in the following table:

Date			Observation
dd	mm	yyyy
20	02	2002
30	02	2003	Not a date
01	02	2010
11	02	2011
21	02	2012
31	02	2013	Not a date
02	02	2020
12	02	2021
22	02	2022
32	02	2023	Not a date
03	02	2030
13	02	2031
23	02	2032
33	02	2033	Not a date
04	02	2040
14	02	2041
24	02	2042
34	02	2043	Not a date
05	02	2050
15	02	2051
25	02	2052
35	02	2053	Not a date
06	02	2060
16	02	2061
26	02	2062
36	02	2063	Not a date
07	02	2070
17	02	2071
27	02	2072
08	02	2080
18	02	2081
28	02	2082
09	02	2090
19	02	2091
29	02	2092	Not a date

Then I searched for palindromic dates in this century in the American way:
mm/dd/20yy

and
mm/02/yyyy

The month may be 01, 02, 03,?, 12:

mm	dd	yyyy
02	02	2020
03	02	2030
04	02	2040
05	02	2050
06	02	2060
07	02	2070
08	02	2080
09	02	2090
11	02	2011
12	02	2021

Days between the Palindromic Date for British & American ways:

For British way, if first year is 20/02/2002:

-19 days + 8 years
+ 10 days + 1year
+ 10 days + 1year
-19 days + 8 years
:
:

For American way, if first year is 01/02/2010:

+ 10 days + 1 year
- 9 days + 9 years
:
:

03/02/2030 then:

+ 1 day + 10 years
:
: