The mate-in-n problem of infinite chess is decidable

@article{Brumleve2012TheMP,
  title={The mate-in-n problem of infinite chess is decidable},
  author={Dan Brumleve and Joel David Hamkins and Philipp Schlicht},
  journal={ArXiv},
  year={2012},
  volume={abs/1201.5597}
}
The mate-in-n problem of infinite chess--chess played on an infinite edgeless board--is the problem of determining whether a designated player can force a win from a given finite position in at most n moves. Although a straightforward formulation of this problem leads to assertions of high arithmetic complexity, with 2n alternating quantifiers, the main theorem of this article nevertheless confirms a conjecture of the second author and C. D. A. Evans by establishing that it is computably… 

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It is proved that every countable ordinal arises as the game value of a position in infinite three-dimensional chess, and consequently the omega one of infiniteThree-dimensional Chess is as large as it can be, namely, true omega one.

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