First I tried starting with the number 10 but found that I needed two 8s to complete the sequence.
Nine needed two 9s!
But 8 was fine!
I wondered if it would work with factors of 8, but 4 needed a 16 to complete the sequence.
When I tried 1 and 2 I spotted that the numbers needed to complete the sequence had a difference of 1.
I decided to create a grid with all the numbers 1 - 15 on the left hand side, ready to work across. I realised that the columns would ascend or descend in patterns of +1 or -1.
Anything that began with a number higher than 10 would require numbers 16 - 20 to complete the sequence.
Anything that began with a number lower than 5 would require numbers 16 - 20 to complete the sequence.
And then it was a case of completing the remaining sequences that began with 5, 6, 7, 8, 9 and 10.
These sequences showed that some numbers failed:
10 - needed two 8's
9 - needed two 9's
7 - needed two 8's
and 5 - needed two 10's
Therefore the only correct sequences could begin with either a 6 or an 8.
I know that I have found all the different solutions, but did not need to create them all as I found out early on that I could predict some sequences would use numbers above 15, because they went up or down in steps of 1.
To be honest, the explaining took longer than the working out of the solution!