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Rook Roundtrip

In the picture, you see the longest roundtrip of a rook such that there are no "shortcuts" - imagine the target squares are given, then at each square you have exactly two other given squares accessible to the rook, and they connect to a round trip. (Trivially the longest rook trip length is 2*8.)

What are the values for the other officers? Q and N are most interesting. (I think I know the maximum for N.)

Note: a1-b1-c1 is forbidden, even if you can't go from a1 to c1 if something stands on b1.

Son of Note: You could formulate this problem far simpler: "Place m queens (knights,...) such that all queens are protected twice". But it's not the same problem anymore - the Qa6 will never "touch" Qh3.

Not a roundtrip

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    Can you explain the details further? Maybe with some small examples.. I couldn't understand what is a "roundtrip".
    – Minot
    Commented Apr 26, 2023 at 6:42
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    @Minot you are looking for a cyclic list of squares such that (i) each pair of consecutive squares is a [queen's] move apart, and (ii) no non-consecutive pair is a [queen's] move apart. Commented Apr 26, 2023 at 8:09
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    Queen looks to be 10 but I can't prove it Commented Apr 26, 2023 at 10:41
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    @AndrewChin: Ditto. (A computer search will take...long :-) Commented Apr 26, 2023 at 18:07

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Here are my attempts for Q and N.

Q record?

Observe that I didn't use g&h at all, maybe 11 are possible after all.

N record?

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