Question

We call a number special if every digit in the number either is a 1 or borders a 1. For example, 11111, 13, 141, 1441, 515151, and 101 are all special, but 10001, 222, 122, and 1333 are not special.

How many 2-digit numbers are special?

Answer 1

2-digits, can't be hard

9 numbers ending in 1
10 numbers beginning with 1

11 occurs in both lists.

Answer 2

Any 2-digit number with a 1 is special (and all special numbers have at least one 1). There are nine 2-digit numbers with ones digit 1, and, there are nine additional 2-digit numbers with tens digit 1 (except from 11, which we've already counted, since its ones digits is 1). Therefore there are 9+9=18 special 2-digit numbers.