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Puzzling

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  • $\begingroup$ I'm not convinced by the argument in your last paragraph. For example, if we ask instead the number of parity under the whole board, then we can still write it as $R_{1, *} + R_{2, *} + \dots + R_{8, *}$. These $8$ vectors (or matrices, in your language) are also linearly independent, thus the last paragraph of your argument seems to imply that the answer for this variation is also $8$. This however is obviously wrong, as the whole board is just one single rectangle. $\endgroup$
    – WhatsUp
    Commented May 15, 2021 at 16:41
  • $\begingroup$ Yes, I see the point and agree. This needs some fixing. $\endgroup$
    – Léreau
    Commented May 15, 2021 at 20:24