The 9s pattern

In your book, Maths Mela (Chapter 1), it is Mira's turn to give Sanju a puzzle. Take any two different digits, say 3 and 7. Make two 2-digit numbers with them: 37 and 73. Subtract the smaller from the bigger: 73 − 37 = 36.

Now use the two digits of the answer, 3 and 6, and do it again: 63 − 36 = 27. Again: 72 − 27 = 45. Again: 54 − 45 = 9. We have reached a 1-digit number, so we stop. Mira says: 'No matter which two digits you choose, you will get 9 in the end.' How did she know?

Look at the answers: 36, 27, 45, 9. Every one of them is in the table of 9. Mira made a table to see why. With digits 3 and 7, the digits differ by 7 − 3 = 4, and 73 − 37 = 36, which is 4 × 9. With digits 1 and 9, they differ by 8, and 91 − 19 = 72 = 8 × 9. With 2 and 8, they differ by 6, and 82 − 28 = 54 = 6 × 9. With 4 and 5, they differ by 1, and 54 − 45 = 9 = 1 × 9.

So the difference of the two numbers is always 9 times the difference of the digits. That is why the answers stay in the table of 9, and why the puzzle always ends at 9. If the digits differ by just 1 (like 4 and 5, or 6 and 7), you get 9 at the very first step. To get 27, which is 3 × 9, choose digits that differ by 3, like 1 and 4: 41 − 14 = 27.