Fifteen cards
The 1st and 2nd cards must add up to 15, so the combinations can be:
1-14, 2-13, 3-12, 4-11, 5-10, 6-9, 7-8
The 2nd and 3rd card must add up to 20, so the combinations must be:
5-15, 6-14, 7-13, 8-12, 9-11
The 3rd and 4th card must add up to 23, so the combinations must be:
8-15, 9-14, 10-13, 11-12
The 4th and 5th card must add up to 16, so the combinations must be:
1-15, 2-14, 3-13, 4-12, 5-11, 6-10, 7-9
The 5th and 6th card must add up to 18, so the combinations must be:
3-15, 4-14, 5-13, 6-12, 7-11, 8-10
The 6th and 7th card must add up to 21, so the combinations must be:
6-15, 7-14, 8-13, 9-12, 10-11
My cards could be:
6,9,11,12,4,14,7
9,6,14,9,5,13,8
When I went through all the numbers on the first one and none of them worked unless I started on 6 or 9. The first number has an effect on the whole sequence, as the second number is dictated and so on and so forth. This means that there are only 15 different attempts, of which only 2 is correct.