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MatPlus.Net Forum General Most captures on a single square in a PG?
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(1) Posted by Rewan Demontay (Real Name: James Malcom) [Wednesday, Oct 28, 2020 01:56]

Most captures on a single square in a PG?

I was browsing PDB today and came across P1338395-, which shows a nice little double knight CF.

Andriy Frolkin & Sergej Tkatschenko, J.Lois-70 JT 2016-2017, Lob
PG in 13.0
(= 14+10 )

Henrik Juel's comment noted the "heavy action on d8," aka there were four captures on that one square. This elicited today's question: What is the highest body count on a single square shown to date in a PG? Now, the most that is possible in a non-unique PG is 19 without promotions, and 26 with, as the great Bader Al-Hajiri showed long ago on Krabbe's site: I don't know of any way to feasibly search for such a PG in PDB, and massacre games are of no use here, hence I must ask the MatPlus general populace.

Showing 4 captures is actually pretty easy to do, as it turned out, in a short manner.

PG 4.5
(= 14+14 )

1. b4 a5 2. Ba3 axb4 3. Bxb4 Ra4 4. a3 Rxb4 5. axb4

Both White and Black capture on b4 together 4 times in total.
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(2) Posted by Hauke Reddmann [Wednesday, Oct 28, 2020 11:42]

The next step, of course, would be N consecutive captures :-)
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(3) Posted by Jakob Leck [Wednesday, Oct 28, 2020 15:26]

STIP='BP' AND NOT G='Fairies' AND SOL='xb4%xb4%xb4%xb4'
Modify appropriately and repeat 63 times. ;)
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(4) Posted by Andrew Buchanan [Wednesday, Oct 28, 2020 17:02]

I think Al-Hajiri was missing a trick. Without promotions, you can have 20 captures on one square (supported by 3 captures on other squares, leaving 9 pieces on the board) in a non-unique game. 1. d4 c5 2. c3 cxd4 3. cxd4 e5 4. e3 exd4 5. exd4 Nc6 6. Bb5 Nxd4 7. Bc6 bxc6 8. Qxd4 d5 9. f3 c5 10. Be3 cxd4 11. Bxd4 Bf5 12. Bc3 d4 13. Bxd4 Be4 14. fxe4 f6 15. e5 fxe5 16. Nd2 Qxd4 17. 0-0-0 0-0-0 18. Ndf3 Bc5 19. Rxd4 exd4 20. Nxd4 Nh6 21. Ndf3 Rd4 22. Nxd4 Bxd4 23. Nf3 Rd8 24. Nxd4 Rxd4 25. Rd1 Nf5 26. Kc2 Kd8 27. Kc3 Ke8 28. Rxd4 Nxd4 29. Kxd4 leaves us with:

(= 5+4 )
Maybe you can do it a bit quicker, but I think 20 captures is the limit.

Note this is not really a "non-unique proof game" because there is nothing forcing all these captures to be on the critical square, even in a non-unique way.

This solution can't extend naturally to an optimal solution with promotions. There, the record remains 26. That's easy to see: suppose that you discount the 3 files surrounding the critical square. Then there are 5 files left to clear, maybe divided somehow. But if you clear these with just 3 captures, then 2 of the captures must be of pawns. We have 2 off-colour bishops though. Thus at least 4 supporting non-thematic captures are required, and Al-Hajiri showed this is sufficient.

Now for unique proof games. The shortest number of moves in a unique proof game with N captures on a critical square, for N = 0, 1, 2, 3, 4 I think is f(N) = 0.0, 1.5, 2.0, 3.5, 4.0. The 4.0 is quite pretty for something so short, with dual elimination and tempo. I hope I'm not missing anything shorter!

(= 14+14 )
SPG in 4.0

EDIT: have gone from 30.0 to 28.5 moves for the 20 captures, now with d4 as the critical square
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(5) Posted by Rewan Demontay (Real Name: James Malcom) [Thursday, Oct 29, 2020 02:13]

The great Al-Hajiri being outwitted? Congratulations, but not the first time, Andrew! See #377-

The PG 4.0 with 4 captures on a6 is indeed a rich little nugget-now we await 5 and more!
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(6) Posted by Rewan Demontay (Real Name: James Malcom) [Thursday, Oct 29, 2020 04:48]

Thanks for the suggestion, Jakob-I added on 'AND NOT COOKED AND NOT K='Fairies". However, nothing with 5 captures or more showed up for any of the 64 squares. I found a few more with four captures, so that seems to be the current known record, at least here, so far.
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(7) Posted by Hauke Reddmann [Thursday, Oct 29, 2020 09:19]

(= 14+14 )

SPG 4.5

Not surprisingly, two more captures on a5 are no longer unique,
but maybe you have an idea. (Zoinks! My first SPG ever!)
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(8) Posted by Michel Caillaud [Saturday, Nov 7, 2020 21:08]

This question about such a SPG was raised some years ago by Mario Parrinello.
Since then, I had in my files a position with 6 captures (also on a6) in 11.0 moves with an extra "non thematical" capture.
Reviewing it now, I find the scheme can be done in a neater way :

(= 13+13 )

SPG 9.0 C+
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(9) Posted by Theodoros Giakatis [Saturday, Nov 7, 2020 21:56]

Very nice, ...O-O-O is the idea which save one move for Black.
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(10) Posted by Rewan Demontay (Real Name: James Malcom) [Saturday, Nov 7, 2020 22:24]

Michel, very ingenious! 6 will surely is and ought to be the record. The use of 0-0-0, as Theodoros noted, is the neat little bugger that make it unique. It took Jacobi 479 seconds to provide additional verification.
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(11) Posted by Andrew Buchanan [Sunday, Nov 8, 2020 06:13]

(= 13+13 )
PG in 7.0

Here's my effort at 6 captures on a6. Jacobi C+. For N=0,1,2,3...6 this extends the sequence f(N) to 0.0, 1.5, 2.0, 3.5, 4.0, 6.5, 7.0 showing every intermediate value during the course of the proof game. However maybe shorter proof games are still possible? Or maybe this sequence can be extended to more captures, perhaps again within a single proof game?
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(12) Posted by Michel Caillaud [Sunday, Nov 8, 2020 19:13]

Well done, Andrew!
For this kind of proofgame, Euclide seems the fastest (for the time being...).
I guess that with a low number of moves, François Labelle can explore if more realizations or more capturing moves are possible.
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MatPlus.Net Forum General Most captures on a single square in a PG?