# [Retros] happy prime new year

andrew buchanan andrew at anselan.com
Sun Jan 8 14:24:53 EST 2017

```Hi Francois,
Thanks for your email. I wanted to give others a chance to opine, but I guess it's maybe a bit of a narrow subject!
Congrats on the PG in 3.5, with double switchback.
Your idea of defining "interesting" reminds me about the "interesting number paradox". Basically what you have proposed to do sounds fine, as one approach to this whole unexplored area. And there are other possibilities.

I personally would like to know what the smallest queue problem is for each natural number, and (closely related) what is the smallest linear extension problem. (The difference being that e.g. 1. a4 a6 2. Sa3 a5 3. Sc4 = 1. Sa3 a6 2. Sc4 a5 3. a4 is a queue problem but not a linear extension problem.
I am particularly interested in symmetric diagrams that are smallest queue or smallest linear extension, if the play or the arithmetic is different for each side. Do you have one for 2017?

Alternatively, if there is some difference between the sets of moves, then Noam's approach has been to restrict that just a single move.

Thanks,
All the best,Andrew

On Tuesday, January 3, 2017 10:29 PM, Francois Labelle <flab at wismuth.com> wrote:

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 asymmetric logic), please enjoy:

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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