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(1) Posted by Hauke Reddmann [Thursday, Jan 26, 2023 09:34] 
Longest helpmate? And I don't mean "longest helpmate *correct as a problem* ".
Those values are easily googled (48; 111 if illegal position).**
But rather along this question*
https://chess.stackexchange.com/questions/18587/helpmateofarbitrarylength
 it doesn't matter whether there are several possibilities to reach
the goal, it's just that the distancetomate N under helpplay is maximal.
* With the same intent  I am an arbiter and N>50 (75?) would make epic
trolling material :)
** Never trust anything googled  seems to be Madrasi? YACPDB 409595 
and notation/position won't match...R must be K 

(2) Posted by James Malcom [Saturday, Jan 28, 2023 17:27] 
Do you mean for fairy helpmates?
I bump you to Madrasi h#52: https://pdb.dieschwalbe.de/P1287073 

(3) Posted by Hauke Reddmann [Sunday, Jan 29, 2023 10:24] 
Nope, I exactly mean NOT fairy and was a bit miffed
Google tricked me :) Orthodox, please. And again,
rather "shortest mate line possible down the game tree"
than "helpmate". Illegal position would be
acceptable, but that's a different record. 

(4) Posted by James Malcom [Sunday, Jan 29, 2023 20:49] 
If you mean a game in which both sides help prolong the inevitable but agree to arrange a checkmate, wouldn't it'd just be the longest game possible?
8848.5 moves: https://wismuth.com/chess/longestgame.html 

(5) Posted by Kevin Begley [Monday, Jan 30, 2023 06:20] 
Related question (or is this THE question): What is the longest possible helpmate if you do not require a unique solution?
So, you're looking for a position where there is no helpmate in under 28 moves (you want something longer than Bernhard Hegermann's record), but you care nothing about whether the solution is unique.
In other words, what's the longest number of moves for a Shortest Helpmate problem (akin to Shortest Proofgame problems, where the move order and transpositions are of no concern)?
Answers in any of these three varieties:
1) no promoted force and legal position,
2) promoted force and a legal position, and
3) an illegal position (no care about promoted force).
What are the records here?
Has any composer explored these questions at significant depth? 

(6) Posted by Joost de Heer [Monday, Jan 30, 2023 08:59] 
Or in mathematical terms:
Let S be the set of nondead positions (including illegal positions).
Let M(P) be the minimal number of halfmoves needed to reach a mate (any king, so black mating white is also fine) in a position P from this set.
Obviously, the minimum of M is 0 (all positions that are already mate), but what is the maximum?
Examples:
* M(P) of the Vielvaeterproblem is 1 if white has the move, and 4 if black has the move. In both cases the sequence is unique.
* M(P) of the start position is 4 (1. g4 e5 2. f4 Qh4#), in this case the sequence is not unique.
* For any random position in the set: Try to find the shortest helpmate, mating the black king, and the shortest helpmate, mating the white king. M(P) is the minimum of these two (use infinity for when it's not possible to mate the black or the white king. Since the position isn't dead, it's not possible for both values to be infinity)
A related problem is to find max(M(P)) for any position P in the set S' of legal nondead positions. 

(7) Posted by Kevin Begley [Monday, Jan 30, 2023 09:36] 
Well said, Joost. 

(8) Posted by James Malcom [Monday, Jan 30, 2023 09:43] 
Here are your answers, Kevin.
1) Ottó Titusz Bláthy, 56 Magyar Sakkvilag 01/02/1924, Helpmate In 40
(= 8+10 )
https://pdb.dieschwalbe.de/P1379073
2) Karl Fabel, Schachmatt 1948, Helpmate In 48
(= 11+10 )
https://pdb.dieschwalbe.de/P0569740
3) James Malcom & Arno Tüngler, Matplus.net Forum 1/30/2020, Helpmate In 111
(= 14+31 )
https://pdb.dieschwalbe.de/P1372555 

(9) Posted by Kevin Begley [Monday, Jan 30, 2023 10:15] 
Wow. Thank you, James! 

(10) Posted by Hauke Reddmann [Monday, Jan 30, 2023 10:32] 
THX, Malcolm. Fabel missed the 50 by >< that much
(the 111 will probably not occur OTB that soon :)).
Indeed, Kevin and Joost's reformulations were
my intent. That the h48 is not completely exact
is irrelevant for OTB trollolo purposes :) 

(11) Posted by Ulrich Voigt [Monday, Jan 30, 2023 11:07] 
Note that both the h#40 and the h#48 feature a capture long before the end, so if you intend to do some trolling via the 50 move rule, you are much farther from your goal than you might have hoped. 

(12) Posted by Joost de Heer [Monday, Jan 30, 2023 11:27] 
 

(13) Posted by Joost de Heer [Monday, Jan 30, 2023 11:30] 
By the way, has anyone checked these composition to see whether black can mate white at an earlier stage? Because that would decrease M(P) as it measures the distance to mate, not the distance to mate of the black king. 

(14) Posted by Viktoras Paliulionis [Monday, Jan 30, 2023 18:39] 
In the position of Malcom & Tüngler, White can be mated in 107.5 moves. 

(15) Posted by Sarah Hornecker [Monday, Jan 30, 2023 20:57] 
There is no magical switch however that turns the diagram from Black to play to White to play. So that isn't a cook.
EDIT: Wait, you said "White can be mated". So it would be kind of a Duplex? 

(16) Posted by James Malcom [Monday, Jan 30, 2023 21:08] 
As I understand it, we are looking for positions, under Kevin's system, that have the longest shortest possible help play for either side to be mated. Neither side has any intention to prolong.
The positions I posted are the longest with Black specifically to be matednot quite the goal, but a starting to point to adapt from.
So, from that stance, unless we can prove which side is to move, as in the h#111 that is a h#107.5 for this goal, we mjst find the shortest possible mates in the h#40 & h#48 positions, as are starting points for this challenge. 

(17) Posted by Viktoras Paliulionis [Monday, Jan 30, 2023 21:08] 
It would be a duplex if the numbers of moves match. 

(18) Posted by Joost de Heer [Monday, Jan 30, 2023 22:02] 
You're still looking at the helpmate that is required to show that a position is not dead as a composition. For the determination whether a position is dead or not it doesn't matter which king is mated, as long as at least one king can be mated. 

(19) Posted by James Malcom [Monday, Jan 30, 2023 22:46] 
To whom do you speak, Joost?
To Viktoras, I do wonder what the duplex record would be, both sides having the same number of moves... 

(20) Posted by Kevin Begley [Monday, Jan 30, 2023 23:32] 
My question was poorly conveyed  I was primarily interested to know the record for a given side to move (that has been answered).
The duplex question is interesting, but hardly the droids I'm looking for. 

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