A Combinatorial Problem with a Fibonacci Solution

For many years I have enjoyed lecturing to groups of high school students about the excitement of mathematics. One diversion that never failed to capture their attention was as follows: Everyone was asked to write down a three digit number (abc with a > c), reverse it (to form cba), find the difference (as a three-digit number) between the two numbers, and add the difference to its own reverse. The amazed looks on the students' faces at discovering they had all reached the same end number 1089 was a sight to behold. The elementary algebra