# [Retros] happy prime new year

Noam Elkies elkies at math.harvard.edu
Sat Jan 28 22:23:17 EST 2017

```Francois Labelle <flab at wismuth.com> writes:

> 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 [...]

> It's quite easy to write a program to detect queue problems [...]

> 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

Neat.  I guess one follow-up question (after extending this past 20)
is to study the positions with subtractions such as -5, -8, -16 and see
if there are any new mechanisms to be found to add to the human composers'
arsenal.

> 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

Looks nice.  "No check protection" -- do you then run the same
computation twice, once normally and the other time allowing checks
to be ignored, and then extract the positions that have the same
lengths and enumerations either way?

> 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).

Possibly even closer to my non-SPG twins in 8.5 moves
<http://www.math.harvard.edu/~elkies/2016.html>.

> 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.

Looking forward to your announcements of further discoveries,
--NDE
```