The smallest two-digit number that is not the sum of three different one-digit numbers is

The smallest two-digit number that is not the sum of three different one-digit numbers is 12.

Here's the step-by-step explanation:

1. Start with the smallest two-digit number, which is 10.
2. Check if it can be expressed as the sum of three different one-digit numbers. The smallest three different one-digit numbers are 1, 2, and 3, and their sum is 6, which is less than 10. So, 10 can be expressed as the sum of three different one-digit numbers (for example, 1+2+7, 1+3+6, 1+4+5, etc.).
3. Move to the next two-digit number, which is 11. Again, check if it can be expressed as the sum of three different one-digit numbers. It can be (for example, 1+2+8, 1+3+7, 1+4+6, etc.).
4. Move to the next two-digit number, which is 12. Check if it can be expressed as the sum of three different one-digit numbers. It cannot be, because the maximum sum of three different one-digit numbers is 1+2+9=12, and in this case, the numbers are not different. So, 12 is the smallest two-digit number that cannot be expressed as the sum of three different one-digit numbers.