I thought of two two-digit numbers, which do not have any digits in common, one being double the other. Moreover, the digits of the smaller number are the sum and the difference of the digits of the larger number. What are the two numbers?
From the outset, we know that the larger of the two numbers

- is even, being double the smaller, but does not end in zero, because the larger and the smaller number do not share any digits;
- its tens digit is at least 2, because its half is a two-digit number, and
- the sum of its digits is at most 9, because this sum is one of the digits of the smaller number.

Therefore, the smallest candidate for the larger of the two numbers is 22, and the largest is 72. After 22, the next even number is 24, which we discard because its half is 12, which repeats the digit 2. Already 26 is a candidate by this criterion, but 28 is not, as the sum of its digits equals 10. Continuing up to 72, we obtain all the candidates, as indicated in the following table.

| larger | 22 | 26 | 32 | 34 | 36 | 44 | 54 | 62 | 72 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| smaller | 11 | 13 | 16 | 17 | 18 | 22 | 27 | 31 | 36 |

By verification, 34 and 17 is the only solution, being the two numbers I thought of.