Let’s still muse a bit about the labyrinth the algorithm faces. In this post’s illustrations, green points are as before mandatory, given by the current state of affairs. Black points are possible partial sums without respect to permutations. Cyan point fillings are partial sums appearing to pass the enforced restrictions. As red circles as before point to partial sums which must be reached on that particular row, it is good, that there is no red point without cyan filling. The next good thing is: Only on rows with red circles there are black points without cyan filling left – the restrictions do work, but there are still too few restrictions to be efficient.
Of course one has to balance between the costs of uneducated searches vs. the costs of strict restriction computations …