Building on what Jeff and Car have said, the problem reduces to finding three digits that are the sum of two smaller digits, while none of the digits are repeated.

These are the possible sums of digits where each digit on the left of the equal sign is the sum of one of the pairs on the right side of the equal sign:

3 = (1,2)

4 = (1,3)

5 = (1,4), (2,3)

6 = (1,5), (2,4)

7 = (1,6), (2,5), (3,4)

8 = (1,7), (2,6), (3,5)

9 = (1,8), (2,7), (3,6), (4,5)

Now, which digits can be sums?

For any digit that is a sum, none of the digits in one of its pair(s) can be in any other sum's pair(s), and neither can the candidate sum digit be used.

IF "3" is a sum, then neither "1" nor "2" nor "3" can be in any other sum's pair(s), which eliminates "4", "5", "6", "7", and "8", so only "9" could be another sum. This means only two digits can be sums, but we need three. Therefore, "3" cannot be a sum.

IF "4" is a sum, this eliminates "5" and "6", ("3" has already been eliminated). "7" has been limited to (2,5) and using those digits eliminates "8" and "9", so again only two digits can be sums. Therefore, "4" cannot be a sum.

IF "5" is a sum, then using (1,4) eliminates "6" and "7" and limits "8" to (2,6). But using (2,6) eliminates "9", so again only two digits can be sums. Using (2,3) instead eliminates "6", but "7", "8", and "9" would all require the digit "1" as an addend. Therefore, "5" cannot be a sum.

IF "6" is a sum, then using (1,5) eliminates "8", but reduces "9" to (2,7) which means "7" cannot be a sum, which leaves us with only two sums. Using (2,4) instead eliminates "7" and reduces "9" to (1,8) meaning "8" cannot be a sum so we are STILL left with two sums. Therefore, "6" cannot be a sum.

Okay, last chance. If "7" cannot be a sum, then we will never find three sums. Using (1,6) reduces both "8" and "9" to requiring "5" as an addend. Using (2,5) eliminates "8" and we can't take any more eliminations. Using (3,4) reduces "9" to (1,8) which eliminates "8" as a sum. Therefore, "7" cannot be a sum.

That's it, we're done. The problem, unfortunately, has no solution.