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, which sum up to 6. So, 10 can be expressed as the sum of 1, 2, and 7 or 1, 3, and 6 or 1, 4, and 5.
3. Move to the next two-digit number, which is 11. It can be expressed as the sum of 1, 2, and 8 or 1, 3, and 7 or 1, 4, and 6 or 2, 3, and 6.
4. The next two-digit number is 12. Try to express it as the sum of three different one-digit numbers. The combinations of 1, 2, and 9 or 1, 3, and 8 or 1, 4, and 7 or 2, 3, and 7 or 2, 4, and 6 all exceed 12. Therefore, 12 cannot be expressed as the sum of three different one-digit numbers.
5. So, the smallest two-digit number that is not the sum of three different one-digit numbers is 12.