# [Retros] happy prime new year

Francois Labelle flab at wismuth.com
Sat Jan 28 13:45:59 EST 2017

```andrew buchanan wrote:
> 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.

Hi Andrew,

It's quite easy to write a program to detect queue problems (for those
who haven't seen the papers, let me copy the definition: "a queue
problem is a chess problem in which each solution has the same set of
moves, but the order of the moves can vary").

Is this what you want?

1: 0.0 moves
2: 1.5 moves 1. b4 Nc6 2. d4                 2 = 2*1
3: 2.5 moves 1. b4 a5 2. d4 Ra6 3. Bh6       3 = 3*1
4: 2.0 moves 1. b4 Nc6 2. d4 Nf6             4 = 2*2
5: 2.5 moves 1. b4 a5 2. e4 Nc6 3. Ba6       5 = 3*2 - 1
6: 2.5 moves 1. b4 a5 2. d4 Ra6 3. f4        6 = 6*1
7: 3.0 moves 1. b4 e6 2. d4 Nc6 3. Bh6 Qg5   7 = 3*3 - 2
8: 3.0 moves 1. b4 a5 2. e4 Ra7 3. Ba6 Nf6   8 = 3*3 - 1
9: 3.0 moves 1. b4 a5 2. d4 Ra6 3. Bh6 Nf6   9 = 3*3
10: 3.0 moves 1. b4 a5 2. e4 Nc6 3. Ba6 Nf6         10 = 3*6 - 8
11: 3.5 moves 1. c4 a5 2. e4 Ra7 3. c5 Nf6 4. Ba6   11 = 4*3 - 1
12: 2.5 moves 1. b4 Nc6 2. d4 Nf6 3. f4             12 = 6*2
13: 3.5 moves 1. d4 a5 2. e4 h5 3. Bh6 Ra7 4. Ba6   13 = 6*3 - 5
14: 3.5 moves 1. d4 a5 2. e4 Ra7 3. Be3 Nf6 4. Ba6  14 = 5*3 - 1
15: 3.5 moves 1. d4 a5 2. Bh6 Ra6 3. e4 Nf6 4. Bd3  15 = 5*3
16: 3.0 moves 1. a4 e6 2. c4 Nc6 3. e4 Ba3          16 = 6*3 - 2
17: 3.5 moves 1. d4 a5 2. Bh6 Ra7 3. e4 Nf6 4. Ba6  17 = 6*3 - 1
18: 3.0 moves 1. b4 a5 2. d4 Ra6 3. f4 Nf6          18 = 6*3
19: 3.5 moves 1. a4 e6 2. Ra2 Qh4 3. g3 Ba3 4. f4   19 = 12*2 - 5
20: 3.5 moves 1. d4 e6 2. f3 Nc6 3. g3 Qh4 4. Bg5   20 = 12*3 - 16

For each number I picked a shortest example, and in each case I was also
able to choose a captureless example. My computer only checked for queue
problems, but it looks like most of the examples are also linear
extension problems. Checking manually, I think only White's play in #11
is not a poset.

I found a few ways to get 2017 in 7.0 moves with a symmetric diagram.
One example has no check protection and is particularly clean:

http://www.janko.at/Retros/d.php?ff=r3kbnQ/ppp1ppp1/8/3N1b2/3n1B2/8/PPP1PPP1/R3KBNq

Partial order of moves for White are

Nc3 < Nxd5
d4 < Bf4
"  < Qd3 < Qxh7 < Qxh8

Mathematically we can write
(a1,a2,a3,a4,a5,a6,a7) in S_7 (ordering of white moves)
(b1,b2,b3,b4,b5,b6,b7) in S_7 (ordering of black moves)

with constraints for White (similarly for Black):
a1 < a2
a3 < a4
a3 < a5 < a6 < a7

and cross-color constraints:
a3 <= b2 and b3 < a2
a6 <= b4 and b6 < a4

The above mathematical problem is neat because its description is short
and it has 2017 solutions, but that doesn't necessarily make it easy to
solve humanly and in this case I think that it's hard to count solutions.

> 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?

Here's a non-shortest PG in 6.5 moves for 2016:

http://www.janko.at/Retros/d.php?ff=2bqkbnr/pr1npppp/1p1p4/2p5/2P5/1P1P4/PR1NPPPP/2BQKBNR

2016 = 84*24 -- no interaction at all between White and Black. This is
similar to your 2016 = 72*28 in 8.5 moves from last year (except that
yours is a SPG).

I haven't found an example for 2017 yet. Adding the requirement of a
*shortest* PG would make the search even harder since I'm looking at
mirror symmetry and not rotational symmetry.

François
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