[Retros] happy prime new year

[andrew buchanan]

Hi Francois,
Thanks for all your work.
I agree that the 1-sided approach is the best way in. I am particularly happy that you have picked up the distinction between posets and queue problems. 

There would be a few more steps necessary to make sure we can get every number from 1 to say 2500 in non-interactive 2-sided games, ideally with a canonical solution:
(a) find 1-sided solutions for every prime number from 53 to 2477.
(b) guarantee that any two 1-sided solutions are compatible.    (b.1) make sure that representative solutions do not go beyond 4th rank. (e.g. instead of 1.b4 2.d4 3.Ba6 have 3.Ba3). (And given that "longest moves" cannot be a principle of canonicality, let's go with "shortest moves" instead. Hence maybe make *all* moves as short as possible, so: 1.b3 2.d3 3.Bd2). Actually where possible, representative solutions should not go beyond the 3rd rank, because eventually we fill find some numbers can only be attained minimally by solutions that go beyond 4th rank, e.g. to 5th rank.
    (b.2) make sure that the king is not exposed to possible diagonal checks. So 1.b3 2.e3 3.Be2 instead. (I guess castling is not an issue for any case you have found so far?)
    (b.3) Interesting one: compatibility of length of moves. As long as the two sets of moves are within 1 of each other in terms of length, then we are set. But we can't at the moment even combine the given 1-sided solutions for 1 (length=0) and 2 (length=2). So we need alternate lists for different lengths. To give the *shortest* non-interactive game we need 1-sided solution of length k* for all k<k*<7. And we also need to know what the maximum length is of solutions for prime n from 53 to 2477 - looking at your OEIS post up to 1000, length slowly grows to 9. (length 10 is only reached for n=979, which is composite).(c) handling unwieldy cases. We may for bigger n find that the shortest solution *must* move beyond the 4th rank, to 5th rank and beyond. For small n, there is so much flexibility (e.g. 1.a3 2.b3 versus 1.e3 2.f3 etc), but others might need to be handled on a case by case basis. At least there can be automation up to a point.
(d) A more open-ended question is whether we can ensure through this that we get the shortest solution that does not involve interaction between the two sides. In principle, to derive a solution for n, we need to consider all a=<b such that ab=n and have min(max(length(a),length(b)), etc. 

(e) A final optional step towards canonicality would be some kind of "alphabet chess"-like idea. Perhaps the lexicographically earliest candidate sequence of move squares consistent with the above will be taken as the representative. So 1.a3 2.Ra2 3.b3. Maybe Black moves can be taken with a reversed ordering of letters, and also numbers, so 1.h6 2.Rh7 3.g6, because this may help give the greatest compatibility between different sides, as we find that minimal solutions cannot be kept within the first 4 ranks.

Hope this helps,Andrew
   
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