A perfect solve, manxman!
4 and 5 must be at the ends since the sum of 1 through 9 is 45
4.......5
The sum of all numbers between and including 4 and 3 is 34. Add the 5 at the other end for 39 and that leaves 6 (45-39). It must BE 6, not 1 and 5 or 2 and 4, because the 4 and 5 are already claimed
4.....365
And keep going.