You may wish to try Always a Multiple? and Special Numbers before tackling this problem.
Where should you start, if you want to finish back where you started?
Alison chose a two-digit number, divided it by $2$, multiplied the answer by $9$, and then reversed the digits.
Her answer was the same as her original number!
Can you find the number she chose?
Then she chose another two-digit number, added $1$, divided the answer by $2$, and then reversed the digits.
Again, her answer was the same as her original number!
Can you find the number she chose this time?
Charlie chose a two-digit number, subtracted $2$, divided the answer by $2$, and then reversed the digits.
His answer was the same as his original number!
What was Charlie's number?
Choose a number, subtract $10$, divide by $2$ and reverse the digits.
What number should you start with so that you finish with your original number?
Choose a 3-digit number where the last two digits sum to the first (e.g. $615$).
Rotate the digits one place, so the first digit becomes the last (so for the example, we get $156$).
Subtract the smallest number from the largest and divide by $9$ (which is always possible).
What do you notice about the result? Can you explain why?
With thanks to Don Steward, whose ideas formed the basis of this problem.