A teen-number pattern

In your book, Maths Mela (Chapter 13, The Transport Museum), there are four multiplications with their answers: 12 × 13 = 156, 11 × 14 = 154, 13 × 13 = 169 and 11 × 12 = 132. Some digits are coloured. Can you see the pattern?

Look at 12 × 13 = 156. The answer starts with 1. The middle digit, 5, is 2 + 3, the two ones digits added. The last digit, 6, is 2 × 3, the two ones digits multiplied. Check 11 × 14 = 154: 1 + 4 = 5 and 1 × 4 = 4. And 13 × 13 = 169: 3 + 3 = 6 and 3 × 3 = 9.

Why does it work? Think of the grid. For 12 × 13, the parts are 10 × 10 = 100, then 10 × 3 and 2 × 10, which make (2 + 3) tens = 50, and last 2 × 3 = 6. So 100 + 50 + 6 = 156.

The pattern works only while the sum AND the product of the ones digits are both single digits. 12 × 14 = 168 and 11 × 18 = 198 follow it. But for 13 × 14, the product 3 × 4 = 12 has two digits, and the real answer is 182. For 11 × 19, the sum 1 + 9 = 10 has two digits, and the real answer is 209.