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Department of mathematics and Informatics, Vilnius University, Institute of Computer Science
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Cycles are eliminated from $n \times n$ boards with precisely $\lceil (n^2 - n/2 + 4/3) \rceil - \lceil n/2 \rceil$ removed cells, a result that is $\lceil((n-1)^2+1)/3\rceil$ or one greater depending on $n \pmod 6$.