This problem offers an interesting and challenging exercise in place value. It can be solved using an experimental approach or more formally using algebra. The task offers rich extension possibilities exploring a context for 'clock arithmetic'.
Ask students to work in pairs to decide what the two meters will read after 1 mile, 10 miles, and 100 miles, and then give them time to consider the first part of the problem. There is likely to be some confusion because of the presence of 1/10ths of a mile on just one of the meters. Some students could be asked to feedback on what was tricky/hard to explain/ hard to agree on.
In order to establish a good approach to these questions, aska few different pairs to demonstrate their first ideas/full reasoning at the board. Then ask all students to derive the answer (4953) writing it out clearly, before making up their own initial meter readings and calculating how far they must go before the meters match.
On the board putthe headings "these pairs of meters will never match" and "these will match". Students can record pairs of initial values for their peers to check. At some stage you'll probably need to declare the no-matches list, closed. Students who feel stuck could take a little break to test these 'solutions' and to try to observe what is making them work.
Some students might like to try to develop an algebraic expression for the distances. How can the question be rephrased as an equation?
You could start students off with a simpler question, e.g. if the trip meter registered 000.0 miles and the milometer registered 00009 miles or 00234 miles, how many miles would the car have to travel for the digits to work? When working on the starting question (4631 etc) allow a lot of time for trial and error solutions, encouraging paired discussions on how to make a better trial each time.
Encourage students to lay out the readings from the two dials in place-value columns and to work one step at a time, recording each new reading in turn.