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?

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.

To find out how many 2-digit numbers are special, we need to list all the possible 2-digit numbers and check if they meet the criteria of being special.

The possible 2-digit numbers range from 10 to 99.

Let's start with the tens digit being 1:

If the tens digit is 1, then all the numbers from 10 to 19 are special since they have a 1 in the tens place.

So, there are 10 numbers where the tens digit is 1.

For the tens digits other than 1 (2 to 9), let's analyze the units digits:

If the units digit is 1, then the number will be special regardless of the tens digit. So, there are 9 special numbers for each tens digit other than 1.

Since there are 8 tens digits other than 1 (2 to 9), the total number of special numbers with a non-1 tens digit will be 8 * 9 = 72.

Adding the two cases together:

10 + 72 = 82

Therefore, there are 82 2-digit numbers that are special.

To find the number of 2-digit numbers that are special, we can systematically go through all the possible 2-digit numbers and check if they meet the given criteria.

Let's start by listing all the possible 2-digit numbers, which range from 10 to 99.

Next, we can check each number to see if it is special. To determine if a number is special, you need to verify that every digit in the number either is a 1 or borders a 1.

For example, let's take the number 22. The first digit does not meet the criteria because it's not a 1 and does not border a 1. Therefore, we can eliminate 22 from the list of special numbers.

Continue this process for each number in the range from 10 to 99, and count the number of special numbers you find.

Once you have checked every number, the total count will give you the number of 2-digit numbers that are special.

2-digits, can't be hard

9 numbers ending in 1
10 numbers beginning with 1

11 occurs in both lists.