Timeline for Is equality possible?

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Nov 3, 2020 at 16:05 comment added isaacg Good point! I agree, it's unbalanced relative to that region. That makes me believe more strongly that your hypothesis might be right.
Nov 3, 2020 at 15:49 comment added Paul Panzer @isaacg Well done! But still no counterexample I believe because we can fatten the square by one column and one row, so it includes the two 1's closest to the -4.
Nov 3, 2020 at 15:02 comment added isaacg Actually, now that I look into it further, I think that this is a counterexample, even to the disconnected version of unbalancedness. Take a look at this sequence of moves, which I believe shows that this is not unbalanced relative to your square region, or any other region that includes the 4 but no 1s: docs.google.com/spreadsheets/d/…
Nov 3, 2020 at 7:52 comment added Paul Panzer @isaacg That said I wouldn't insist on connectedness. I'd be happy with whichever makes proving the conjecture easier.
Nov 3, 2020 at 7:39 comment added Paul Panzer @isaacg I don't think that's a counterexample: Choose as the two regions a square that reaches from the corner to the -4 excluding the four 1's and the L-shaped complement.
Nov 3, 2020 at 7:04 comment added isaacg I like the idea, but I don't think it quite works. I think the following is a counterexample. On a diagonal, place 1,1,-4,1,1. Fill the rest with 0s. It's possible to balance either pair of ones, but it's not possible to balance both simultaneously - after the first move, it becomes unbalanced. However, if we allow the regions to be disconnected in the definition of unbalanced, I think that fixes it.
Nov 1, 2020 at 3:57 history answered Paul Panzer CC BY-SA 4.0