Great Britain has had this problem multiple times.
AAA nnn - the original scheme, the very first numberplate issued was "A 1".
AA nnnn - ran in parallel when the pool was almost empty.
nnn AAA - replaced the previous scheme some time in the 1960s
AAA nnn A - followed a couple of years later, the final A started off as "A", a year later it was "B" and so on. I think they skipped "I", "O" and maybe "L" and/or "Z".
A nnn AAA - was next, the first "A" denoted the year.
AA nn AAA - is the current system. The nn started out at 00 and denotes March-August of year 2000, then came "50" which meant the following six months. A year later it was "01" and then "51". Nine years later, "10" and then "60".
There's no need to implement something that stupid, Jan-June and July-December would have worked instead.of March-Aug and Sep-Feb.
The Germans have a totally different system, AAA-AA nnnn where the first AAA (can be 1-3 chars) indicates the issuing authority, for example "SD" could be San Diego and Los Angeles could be "L" or "LA". The other two characters and the 1-4 numerics just need to be unique. When cars are scrapped their numberplates become available again.