# [Retros] happy prime new year

Francois Labelle flab at wismuth.com
Mon Jan 2 21:28:45 EST 2017

```Hi Andrew,

You're right, I found your SPG in 6.0 moves in a file dated April 2015,
among 46 other 16+16 symmetric PGs in 6.0 moves with 2016 solutions. :)

For 2017 I propose:
http://www.janko.at/Retros/d.php?ff=rnbqkbnr/pppp1ppp/4p3/8/3P4/6P1/PPP1PP1P/RNBQKBNR
SPG in 1.5 (2 solutions). How many solutions in exactly 3.5 moves?

This problem is similar to my PG from 2004. It is not mathematically
interesting, but it is not intimidating because what's going on is
clear: White and Black each performs one switchback, and the approach is
to count the number of combinations, and then subtract the number of
solutions with conflicting switchbacks.

Manually I get 92 solutions for white, 23 solutions for black, and 99
conflicting solutions among the 92 * 23 = 2116 combinations, for a total
of 2017.

If one can define "interesting" in a way that can be programmed, then
maybe it's possible to strip-mine PGs and then report only the
interesting ones? For example, maybe for 16+16 PGs with 2017 solutions I
can remove the black pieces and count the number w of one-sided white
solutions, similarly for the number b of one-sided black solutions, and
minimize w * b - 2017, the number of conflicting solutions?

Happy New Year,

François

On 01/01/17 12:51 PM, andrew buchanan wrote:
> Happy New Year!
> http://www.janko.at/Retros/d.php?ff=1n2kbnr/1bpp1ppp/q3p3/pp2r1NP/4P1B1/5N2/PPPP1PP1/R1BQ1RK1
> SPG in 10.0. How many solutions? C+
>
> Unlike in 2015 & 2016, I have not discovered a *symmetric* SPG for
> 2017, but while delving, I did find a first symmetric queue problem
> for 2014, and a much shorter symmetric queue problem for 2016 than
> presented a year ago. So if you like decorative symmetry (with
>
> Happy New Year! (oops 3 years too late)
> http://www.janko.at/Retros/d.php?ff=rnb2bn1/pppq1pp1/3p4/P2rp1k1/1K1PR2p/4P3/1PP1QPPP/1NB2BNR
> SPG in 10.0. How many solutions? C+
>
> Happy New Year! (oops 1 year too late)
> http://www.janko.at/Retros/d.php?ff=r1b1k1nr/pppp1ppp/4Bq2/n3p3/N3P3/4bQ2/PPPP1PPP/R1B1K1NR
> SPG in 6.0. How many solutions? C+
>
> Discussion:
>
> I wanted to use 2025 rather than 2016 as a jumping-off point for 2017,
> because subtraction makes it easier to have a problem in which only
> the *order* of the moves varies between solutions. This is Richard
> Stanley's definition of "queue problem" in "Queue Problems Revisited"
> (2005).
>
> Richard wanted the dependencies between moves to be expressible as a
> "partially ordered set" (aka: "poset"). An important subtlety Richard
> does not emphasize in QPR is that not all "queue problems" can be
> completely represented as a poset. For example the series proof games
> 1.a4 2.Sa3 3.Sc4 and 1.Sa3 2.Sc4 3.a4 are the two ways to reach a
> certain position in 3 moves, but there is no poset which captures why
> 1.Sa3 2.a4? 3.Sc4 is not a third solution.
>
> Noam Elkies, in "New Directions in Enumerative Chess Problems" (2005),
> goes beyond the series roots of the genre, introducing proof games,
> helpmates and directmates. However, in *almost* all of the two player
> examples in this paper, he remains within the "poset problem"
> paradigm. In some, only one side is active combinatorially, while the
> other side marks time. In others, he completely decouples white and
> black play, with the total number of solutions as the product of the
> numbers of white and black sub-solutions.
>
> Chess-math problems are hybrids - composers must strike a balance
> between the two parents. Pure poset series problems emphasize the
> mathematical side: they faithfully render in chess form a mathematical
> abstraction. But chessically it can be interesting to relax Richard's
> guidelines sometimes, as Noam often does. The alternation of white &
> black moves simply cannot be represented as a partial ordering.
> Construction of a total number of solutions as a *sum* of two numbers
> generally requires that the set of moves is not fixed (e.g. Noam's
> 2017). Representing a total number of moves which is not a simple
> product of the white and black multiplicands requires interaction
> between the two players (e.g my 2017). All these features are
> chessically interesting.
>
> Yet, with due respect to the incredible work of Francoise Labelle, we
> don't want to just computationally strip-mine the Proof Games to find
> the shortest examples of a given cardinality. Richard emphasized that
> the computation of the number of possible move orders should be
> mathematically interesting. Francois has also exhibited considerable
> restraint in not telling us everything he has dug up. Surely my new
> 2016 has been sitting somewhere on a disk of his for years!
>
> Have a great year, folks!
> Andrew Buchanan
>
>
> On Sunday, January 1, 2017 8:32 AM, Noam Elkies
> <elkies at math.harvard.edu> wrote:
>
>
> It's that time of the year again:
>
> http://www.janko.at/Retros/d.php?ff=1nb2bnr/1ppp1kpp/4p1r1/p4p2/3B4/1PP2Q1P/P2PPPPR/RNq1KBN1
>
> (C+ Popeye 3.41 in less than 1 second)
>
> NDE
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