[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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