This problem strengthens students' understanding of place value and can help them to appreciate the power of algebra for solving number problems.
You may wish to work on the problems Always a Multiple and Special Numbers first.
Introduce the first problem:
"I chose a two-digit number, divided it by 2, multiplied the answer by 9, and then reversed the digits. My answer was the same as my original number. Can you work out what my number was?"
Give students time to think about the problem and discuss it with their neighbour. Then share approaches. If no-one has used algebra, introduce the idea of representing a two-digit number as 10a + b.
Next, give students the other three problems:
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?
These problems can all be solved using similar techniques:
Always a Multiple provides a geometric as well as an algebraic way of thinking about place value related number puzzles, and might be a useful introduction to the ideas used in this problem.