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a-mazing puzzle. I almost have case m=5 figured out :) n=5 should give 86
its really tough problem
That was quite interesting. First I misread the condititions and thaugt m is limited by 6. So I hardcoded some data to solve the problem. When I saw my mistake I replaced the hardoded data by an algorithm which finds these data. But for 7 and 8 I timed out (in a large scale). Then I discovered that I do sparse matrix multiplications, so I optimized my code for it and managed to solve everything for m=7 and some of m=8. The worst case for m=8 needed nearly 20s. Finally I found a further optimization to solve everything n time.
Is there any way for the first tile hit on the bottom row to not be the bottom right?
I think the people who solved this should get together and publish it in some math or algorithm journal. Also the rating might need to be upgraded a little bit to expert or fiendish for example! Of course that makes it all the more satisfying when finally solved! Just to get people thinking: use a recurrence or enumerate state combinations seems to be the question here.