You are given a two-digit number. You can replace one of its digits with the sum of its digits modulo 10. For example, if the starting number is 58 then you can change it to 38 or 53. You can continue changing this number provided that you do not repeat a previously created number. This means if you went from 58 to 53 then you cannot go back to 58. Leading zeroes are not allowed.
What is the longest sequence of numbers you can create if you start with the number 10?
Think about numbers that cannot be reached. Also think which numbers will get you stuck in a cycle quickly. These two things should give you a good upper bound on the maximum score.