Ha I can't believe I came up with this mess below, but missed that one. There were some ideas I tried with my idea, that I'll now try with yours
:
After doing just the first set of numbers, the grid reminded me of a game I had played a long time ago, but I can't remember what it's called now. The game involved drawing lines, starting in each box with a number in, and the number specified how long the line had to be. You had to fill the whole grid without overlapping lines.
Interestingly this first set of numbers seems to have a valid solution. I know my explanation is very bad, but I have attached my solution to the game.
14432
All done in the finest MS paint with the pencil tool  . |
Well done Dizma, you got what we missed for so long