Hunting rabbits...I mean PICs...is a very arduous task, made even more difficult when recommendations have been made. Mike's recommendation includes EEPROM, so lets go with that just in case there turns out to be some reason you do need to save some info between games.
Mike recommended the 18F2620 or the 18F4620.
Here's the first secret you must have been ignoring on the front of the data sheet:
- 2620: 28 pin part
- 4620: 40 pin part
Only the first digit changes to go from a 28 pin part to a 40 pin part.
Stop sputtering - the TQFP package has 44 pins, but 4 of those are nc. It's exactly the same part as the 40 pin DIP package inside.
"But I want the LF version..."
Stop whining. For the baseline series, there is always an LF version.
18LF4620 should be just what you need.
So let the hunt begin. 10 seconds on Octopart and you'll know what's available. Very difficult.
But how can you be sure that's the part you really want? 2 to 4 and add L? Who believes that works? Microchip will show you it is. Note, there isn't a separate spec page for the LF version.