I thought of 2 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 numbers?
From the beginning, we know about the largest number:

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

Therefore, the candidates for the largest and smallest numbers are:

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

By verification, we find that 17 and 34 are the numbers that satisfy the conditions of the problem.