I would create a data-structure consisting of

• a partly filled square
• the list of numbers you need to place

Then you could use a recursive algorithm like the following:

1. select an element from the number-list (*)
2. select an empty position
3. place the element at the position
4. check, if the square is still consistent (all completely filled lines sum up to 175)
• if yes, repeat the step with the new square and the reduced number-list
• if no, restore the previous state (of number-list and square) and try with the next number and/or position

Once you have your basic algorithm, you can optimize. E.g. your selection-algorithms and your consistency-checks. That's when the real fun starts :-) (because you can add "intelligence" instead of pure "brute-force")

Hope this helps! Rata

(*) a basic selection-algorithm could be going from first to last.