The best way to get started with a Hanjie puzzle is to look for the largest numbers first.
That's because there will be the fewest valid combinations to be placed in these cases. For instance, if there are 9 cells in a row and you see a 7 at the end of it, then there are only 3 possible combinations: cells 1-7, 2-8 or 3-9 must be filled in.
Indeed, in any case where the number at the end is more than half the length of the line, you can instantly colour in the cells that must be filled in any valid combination. Thus in the three combinations bove, notice that cells 3 - 7 are coloured in in all three; hence they must be coloured in the final puzzle.
If you have split cells, for instance 4, 4 note that these must contains a gap in the middle - hence in the 9 row or column puzzle you can instantly fill in any 4, 4 as there is only one combination: 1 - 4, 5 blank, 6 - 9.
Once you have gone through and filled in the easy cells, these should then reduce the number of combinations for the smaller numbers. If you have filled in a cell in the above process that is in a row or column with only one filled cell, then you have automatically placed it! If there are two cells, then you know the cell immediately adjacent on the left or right must be filled. Therefore you can cross out all the other cells in that row or column.
The above technique is important to highlight - just as important as colouring squares you think must be filled is to cross off cells that you know cannot contain colour. Whilst obvious at the time, it is impossible to keep track of all these squares in your mind, and through elimination the answer can sometimes reveal itself!
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