The trick, at your table. Ask someone to pick a three-digit number whose first and last digits are different and to keep it secret. Reverse it. Take the smaller from the bigger. Write the answer with three digits, so 99 becomes 099. Reverse that and add. Say "1089" before they finish.

Try it before chapter 6. Collect five differences on a strip of paper, the way Danya did. What do they all have in common? Look at the middle digit, then at the two ends.

The answer. If the number is abc, then abc take away cba is 99 times (a minus c). So the difference is always 099, 198, 297, 396, 495, 594, 693, 792 or 891. Each has 9 in the middle, and its first and last digits add to 9. Adding it to its reverse gives 900 + 180 + 9, which is 1089.

The grown-up layer. The trick turns place value into a short proof. Children often drop the leading zero, and the "broken" 198 is the best way in to see why place matters. ShowMath has the routine; ProofQuest writes the same idea with letters.

For the ride home. Ask: what would happen with a four-digit number? Try one and see.