[Retros] Fastest mate? :-)

[andrew buchanan]

Hi Noam
Thanks for this. Sorry I wasn’t clear.
Why do we need to prove that Black can’t have just moved? (Apart from making the problem eligible for this mailing list :-) ) 
The “fastest mate” stip doesn’t seem to require that. Whoever has the move, the fractional mates are faster. It just happens that we can also show Black didn’t move last, but the logic of the rest doesn’t build on that.
thanksAndrew

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On Monday, June 12, 2023, 9:40 AM, Elkies, Noam <elkies at math.harvard.edu> wrote:

Andrew buchanan <andrew at anselan.com> writes:

> Great stuff!

Thanks!  So, which of the three possibilities
is the smallest fraction of a move? :-)

> For prior art the nearest is:
> https://pdb.dieschwalbe.de/search.jsp?expression=Probid%3D%27P0004856%27

Thank you.  Mario Richter already alerted me to this one
in separate e-mail.

> However this has no retro logic, and there are
> several mates beyond the Valladao - arguably
> Noam’s non-retro representation is nicer as
> the flight square forces only the three thematic solutions.

Thanks!

> There is a typo in the second diagram, wPf5 should be on e6 I think.

Yes, that's right; good catch, and sorry if the error led
some readers on a wild goose chase.  The correct URL is thus

https://www.janko.at/Retros/d.php?ff=3Q4/3P4/2pkP3/2p1p2R/8/7B/7B/R1K5

> Would be good to see the retro logic spelled out for the first one.

I don't think there's much to do.  Black cannot have just moved:
Na5 can't come from b3 (checking Kc1); nor Pc5 from b6
(impossible bP's); nor bK from d5 or e7 (impossible double check).
But the solutions work: if 0-0-0# then Black just played Nb3a5;
if fxe6 ep# then of course Black just played e7-e5, and it's
easy to retract further (bPc5 came from d7); and if d7-d8Q#
then Black just played Ke7d6.

In another e-mail reply, François Perruchaud suggested that
in the retro setting White might be in the process of promoting on f8
(e8 would also work).  I can stop that by adding two random wP's.
Then I saw that White may also be about to complete d7xc8(e8)#;
if that's an issue then I can block these squares, for example

  https://www.janko.at/Retros/d.php?ff=2BQR3/pppP4/2PkPP2/nPp1p3/2P1P3/7P/7B/R1K5

A similar fix for the non-retro setting seems to require
obtrusive force.  But I can always argue that completing d7xc8(e8)#
can't be the solution because it's a larger fraction of a move than
completing d7-d8Q# :-)

NDE
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