I have fifteen cards numbered $1 - 15$.

I put down seven of them on the table in a row.

The numbers on the first two cards add to $15$.
The numbers on the second and third cards add to $20$.
The numbers on the third and fourth cards add to $23$.
The numbers on the fourth and fifth cards add to $16$.
The numbers on the fifth and sixth cards add to $18$.
The numbers on the sixth and seventh cards add to $21$.

What are my cards?

Can you find any other solutions?

How do you know you've found all the different solutions?