Thank you for your solutions. I received the most complete one form Clement Goh (Age 13) of the River Valley High School, Singapore.

The number 2002 is palindromic because it is the same read from left to right as it is read from right to left. There is one date in February this year where the 8 digits are palindromic if the date is written in the British way (dd/mm/yyyy). This is not palindromic written the American way (mm/dd/yyyy). Is there any palindromic date written the American way this year? Find the next occasions when palindromic dates occur written in each way and the number of days between them.

This is a neat little problem and I have seen other similar ones that involve two numbers that have no common factor (relatively prime). If you cannot see the connection - draw two cogs with a relatively prime number of teeth. Colour a tooth on the smaller red and its starting point on the larger cog. Imagine you could rotate the smaller cog and mark the point on the larger cog that the red tooth would reach after each revolution. Now do the same with two cogs with a number of teeth that are not relatively prime.

Can you see it?

Here is Clement's solution

For this question, I made some cogwheels with different numbers of teeth.

I found out that the number of teeth in both cogwheels must not have the same factor.

For example: with 3 and 4 teeth respectively, the cogwheel with 3 teeth will beable to contact every gap on the cogwheel with 4 teeth.

But cog wheels with 3 and 6 teeth respectively will not be able to do so.

Therefore the values of p and q are:

3 : 4
3 : 5
3 : 7
3 : 8
3 : 10
3 : 11
4 : 5
4 : 7
4 : 9
4 : 11
5 : 6
5 : 7
5 : 8
5 : 9
5 : 11
5 : 12
6 : 7
6 : 11
7 : 8
7 : 9
7 : 10
7 : 11
7 : 12
8 : 9
8 : 11
9 : 10
9 : 11
9 : 12