[Retros] happy prime new year

andrew buchanan andrew at anselan.com
Sun Jan 1 12:51:19 EST 2017

Happy New Year!
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)
SPG in 10.0. How many solutions? C+

Happy New Year! (oops 1 year too late)
SPG in 6.0. How many solutions? C+


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:


(C+ Popeye 3.41 in less than 1 second)

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