Turn it round and subtract
An auto's plate reads 52. Reverse it: write the digits backwards, 25. Subtract the smaller from the bigger: 52 − 25 = 27.
- Digits 5 and 2. Number: 52. Reverse: 25.
- 52 − 25 = 27.
- Gap between the digits: 5 − 2 = 3.
- 9 × 3 = 27. Same answer.
A Kaprekar round
Make the biggest and the smallest number from the digits. Biggest minus smallest is one Kaprekar round. For two digits, that is this trick.
What each digit is worth
52 is 5 tens and 2 ones: 50 + 2. Its reverse, 25, is 2 tens and 5 ones: 20 + 5. The digit 5 is worth 50 in 52. In 25, the digit 5 is worth only 5.
Subtract tens and ones apart. The tens give 50 − 5 = 45. The ones give 2 − 20, which is 18 less. So 52 − 25 = 45 − 18. Now 45 = 9 × 5 and 18 = 9 × 2. So 45 − 18 = 9 × (5 − 2) = 27. Any two digits work this way.
The rule
Subtract the smaller number from the bigger one. Find the gap between the digits: the bigger minus the smaller. Multiply that gap by 9. That is the answer, for any two-digit number. For 52, the gap is 5 − 2 = 3, and 9 × 3 = 27.
Do the trick again on each answer. From 27, reverse to 72: 72 − 27 = 45. The answers are 27, 45, 9, 81, 63, then 27 again. Write 9 as 09, so 90 − 09 = 81.
The rule holds for 40 and 88 too. The reverse of 40 is 04, which is 4. 40 − 4 = 36. The gap is 4 − 0 = 4, and 9 × 4 = 36. The reverse of 88 is 88, so 88 − 88 = 0. The gap is 0, and 9 × 0 = 0.
Set two digits. Reverse the number. Subtract.
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