24387

First I played around with cards and I found that the numbers 8, 7, 13, 10, 6, 12 and 9 worked.

Then I made up a table to see if there were any other solutions.

1, 14, 6, 17, X doesn't work
2, 13, 7, 16, X doesn't work
3, 12, 8, 15, 1, 17, X doesn't work
4, 11, 9, 14, 2, 16, X doesn't work
5, 10, 10, X doesn't work
6, 9, 11, 12, 4, 14, 7 WORKS
7, 8, 12, 11, 5, 13, 8, X doesn't work
8, 7, 13, 10, 6, 12, 9, WORKS
9, 6, 14, 9, X doesn't work
10, 5, 15, 8, 8, X doesn't work
11, 4, 16, X doesn't work
12, 3, 17, X doesn't work
13, 2, 18, X doesn't work
14, 1, 19, X doesn't work
15, 0, X doesn't work