Suppose we have an infinite board with a finite number of chess pieces. The question is whether white can checkmate black (without the after 50 moves it is a draw rule). Can you give an explicit position of which you can prove that white can checkmate black if and only if the Collatz conjecture is true?
This problem is motivated by the fact that for my answer to the question Decidability of chess on an infinite board I did not get any votes. I strongly believe that chess is undecidable but the proof seems to involve a lot of designing of chess positions. That is why I thought this problem would be a good warm up as the Collatz conjecture seems to be easily realizable with a "chess automaton".
First, one should design a position for every number n that can check the parity, divide by 2 or go to 3n+1. I guess this should not be that hard. Then one should devise a mechanism that lets black choose this original number n, maybe something like suggested here Checkmate in $\omega$ moves?.
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