Traditional game-playing computer programs, in particular computer chess programs, use a forward-searching algorithm, evaluating each move, possible countermoves, possible counter-countermoves, and so on, to some depth, and then estimating how favorable the various outcomes are, and using that to select a move.
Endgame tablebases reverse that by focusing on the end of the game when few pieces remain. Because the number of possible configurations is then manageable (e.g., in the millions, which computers can handle), one can compute the eventual outcome of the game under perfect play for every possible position. This is done by starting with all positions where a player is checkmated, then going back to positions where someone can deliver checkmate, then going back to positions where someone cannot avoid a position where the other player can deliver checkmate, and so on, till all paths to a win have been found. Thus, an endgame tablebase provides precise, infallible information about the status of every endgame position it covers.
One can ask questions such as, how many moves till a guaranteed checkmate? What response delays checkmate as long as possible? The numbers will be exact. Tablebases have already provided many surprising results, such as showing positions are won which human analysis had surmised were draws, and vice versa.
Tablebases have unveiled winning sequences which are beyond human comprehension. One of the early results in queen pawn vs. queen showed that a series of dozens of seemingly meaningless moves by White's queen brings about an eventual win. As tablebases have grown larger, accommodating more pieces on the board, more such positions have been found; the pieces dance around in a seemingly random fashion, but the computer assures us they are marching incrementally towards a guaranteed win. Some checkmates take hundreds of moves. In 2012, a computer found a position with a mate in a staggering 549 moves. More discoveries no doubt lie ahead.
The fifty-move rule in chess states that if 50 moves (by each player) pass without a piece being captured or a pawn being pushed—so-called "irreversible" moves—either player may claim a draw. This rule is intended to stop "dead games" where one player won't accept a draw but presses on endlessly. E.g., a player in a position that's stronger but not strong enough to win might continue to seek advantage anywhere, hoping for an error by the opponent or even dumb luck, for hundreds of moves.
The fifty-move rule—hereafter, R50—dates to the 16th century, and has become an integral, if not often invoked, part of chess. For lack of a better term, I shall refer to chess without this rule as unbounded chess.
R50 came under pressure as chess theory advanced and discovered positions where a player has a guaranteed mate that takes more than fifty moves. An early example was two knights vs. a pawn, where some positions require as many as 82 reversible moves to mate (up to 115 total). The argument that the attacking player was simply wasting time no longer held, as a checkmate was provably possible, just after many moves.
As such, a list of exemptions was appended to R50, stating that for certain endgame configurations the moves can be extended beyond 50. This list grew as more positions were found. Eventually, it became clear that this was a bottomless pit: there was no limit to how many configurations would be discovered where a player could argue that more moves were needed, even though many move sequences were beyond human ability. Thus, this enterprise was abandoned, the list of exemptions was abolished, and as of today the rule is again 50 moves in all positions. A checkmate that needs more moves is simply not a win. There doesn't seem to be any other direction for R50 to go, so it will probably remain as-is indefinitely.
Some composers of endgame studies regard R50 as a wart, an ugly concession to practice that invalidates otherwise beautiful wins and adulterates chess. I am not so sure, and here I focus on its role in endgames because (1) it is a rule of chess used all over the world, (2) it has a long and established history as part of chess and may be expected to remain, (3) there are good arguments for its existence, and (4) it is in fact theoretically interesting. The last reason is perhaps most operative.
Many tablebases simply ignore R50, and thus provide lines that may not actually win. This is true of most of the long "wins" alluded to above.
Some available tablebases attempt to incorporate R50 in various ways, but in my view none are satisfactory. The most common is to add a "DTZ50" metric, which counts down to an irreversible move. Many implementations, however, do not handle it correctly because they fail to account for which player made the last irreversible move, and so can be off by a half-move. It also doesn't tell you how many moves are needed to mate, just whether a mate is possible. I have seen another proposal, "DTR", that tries to find the smallest n such that there's a win under an "n-move rule", for n a more general value of 50. This is an interesting idea but it would be difficult to compute, and it isn't really applicable to chess today.
I have searched the Internet to see if anyone had done this right, and all I've found was a single person who in a series of posts in a chess forum provided a correct exposition of this problem and a summary of computer results. This work helped clarify these issues for me as well as supplying much interesting data. I'll call this source Kronsteen after the author's username.
Kronsteen has since independently built several of the same larger tables that I did, and we have been comparing results; Kronsteen found a few errors in an earlier draft of this text, though they were not errors in the tablebases.
I wrote my own software to build this type of tablebase correctly. A summary of discoveries and conclusions follows.
The goal in chess is to win. A mate in 40 is as good as a mate in 4. But often people want to know how many moves are needed to checkmate, and tablebases can provide that answer. We assume that first priority each player tries to win, or at least not lose, and second priority the winner tries to checkmate as quickly as possible while the loser tries to delay as long as possible. Thus, saying, "White mates in 30" means that there is a strategy for White such that Black can delay the checkmate for that long, but not any longer. This is called the "distance-to-mate" or DTM (also called "depth-to-mate", with the same abbreviation), although usually DTM refers to unbounded chess.
Many tables instead provide DTC ("distance-to-conversion"), which is the number of moves before the position can be simplified to an easier position, with a capture or a pawn promotion. This value is easier to compute.
DTZ ("distance-to-zero") is the number of moves before a "zeroing" (irreversible) move. However, since it doesn't count "plies" (see next section), it can't distinguish whether one can win by forcing the opponent to capture or push on their 50th move. DTZ50 is the variant that considers a position drawn if every line (e.g., the sequence after zeroing) runs afoul of R50.
(I use the term irreversible imprecisely, since there are other irreversible moves, such as one that gives up the right to castle, which aren't considered by R50.)
DTM50 is the name given to the distance to mate fully heeding R50.
In chess, "one move" usually means a move by both White and Black, so the term half-move or ply is used to refer to a single move by one player. I will use the latter term here. Thus, R50 really says that a draw can be claimed after 100 consecutive plies which are not a capture, a pawn move, or checkmate.
A consequence of this rule is that if you're shown a chess position, it is not enough to know where the pieces are and whose move it is; you also need to know how many plies it has been since the last irreversible move, what I shall call the ply count, or PC. This extra information is "invisible", much like whether a player still has the right to castle—and can be just as important.
PC can greatly affect the game. For instance, if PC is too high, an otherwise won position may not be winnable as there isn't "time" to exercise the winning moves. Less drastically, for higher values of PC a player may need to change strategy in order to win, and the defending player may be able to use a different strategy to delay the win. In general, if PC is higher, the DTM50 may increase, until for high enough PC the game might be drawn.
The first example on the right is a simple illustration of these concepts as well as the importance of counting plies rather than moves. White has an easy mate in two: 1. Rc7+ Kd8 2. Qc8#. However, if this position is reached after White fumbles about to the point that PC=99, the game becomes drawn: White needs a mate in one, and there isn't one, so after White's next move Black will claim a draw by R50. White needs PC at most 97 to play this line; if PC=97 exactly, White makes it just under the wire, mating on the 100th ply.
What about PC=98? It turns out White can still win, but by a completely different strategy: sacrificing the queen! After 1. Qe6+!, Black is forced to take 1. ... Kxe6, thus resetting PC to 0 in the nick of time. The king rook vs. king endgame is a basic win, in this case a mate in 16. Thus if PC=98, the original position is a mate in 17. The high PC has "cost" White fifteen moves.
These effects can arise even in the simple King Pawn vs. King position. The example on the right is the only instance of this where four different mate lengths arise. White's fastest strategy (for unbounded chess, or low PC) is to move the king in front of the pawn and escort it to promotion, which gives a mate in 17. However, for high PC, White must push the pawn awkwardly early in order to reset the counter. Specifically, for PC 92 or 93, it's a mate in 18; for 94 or 95, a mate in 21; and for 96 to 99, where White must immediately play 1. c3 (not 1. c4?, which is a draw), it's a mate in 22. Kronsteen has the details.
If we accommodate R50, there become 100 times as many possible positions as without it, which obviously greatly increases the resources needed to construct a tablebase. Fortunately, the information is compressible, as the range of possible outcomes is constrained: a series of PC intervals with increasing distances to mate, till PC reaches 99 or it's a draw. This helps, but these tables are still going to be much larger than regular tablebases.
Note that even if we weren't interested in the value of positions with high PC, in theory we will need them anyway in order to count back to find what paths actually win.
I have written software (in Haskell) run on my laptop that has built DTM50 tablebases for a large spectrum of configurations with as many as six men. I will summarize some findings in the sections after next.
Larger tables will require more powerful machines and/or more developed software.
I want to pause here to address an objection to this research. One might protest that the positions being looked at are unrealistic, requiring that one player have squandered dozens of moves doing nothing useful. I have four answers to this:
There are several recurring "tropes" in DTM50 analysis. What follows are a few general categories.
Many endgames consist of a more powerful piece versus a weaker one. The stronger side may first take the weaker piece then checkmate the king, or may even checkmate without bothering with the weaker piece. However, if the PC gets too high, it may be necessary to more rapidly take the weaker piece even if it hurts the overall position.
The first example is an easy illustration. It involves equal material, but White is clearly better-positioned, as, well, the Black queen can be immediately taken. White can in fact mate in two for PC up to 97 completely ignoring the other queen: 1. Qc3+ Kb1 2. Qb2#. But if PC is 98 or 99, White must grab the queen immediately, then mate: 1. Qxh6 Kb1 2. Qd2 (say) Ka1 3. Qd1#. A mate in 2 becomes a mate in 3.
The second position is a rook vs. knight game. These are usually draws, but some are wins, as this one is (for low enough PC!). After 1. Kb3 Ne3, the rook cannot immediately checkmate, but it can go around the knight on the g or h files, and there is nothing black can do but, in the end, throw the knight between the rook and king so it can be taken with checkmate. The position is a mate in 4. If, that is, PC is not more than 93.
For PC=94 or 95, because of the knight's last-gasp suicide, there is no time for this checkmate. Instead, the rook must move to take the knight more quickly. The line is 1. Rf1+ Kb2 2. Rf2+ Kc3 3. Rxg2 Kd4. The knight was taken on the 5th ply, but meanwhile Black managed to run his king to the center, greatly prolonging the win. This king rook vs. king position is a mate in 15, so for these values of PC the original position is a mate in 18, a difference of 14 moves. For PC=96 and above, it is a draw.
The third position is a mate in 9 for PC up to 83. The white king marches across the board while the queen keeps the black king penned in. They work around the bishop and don't need to capture it to checkmate.
However, for PC=84 or 85, White must instead go after the bishop immediately. It can be captured in only 8 moves (one better than 9), while again Black uses the opportunity to position his king better, so that 9 further moves, or 17 in all, are needed for checkmate. The main line is 1. Qb7+ Kg1 2. Qf3 Be5+ 3. Kb1 Bh2 4. Ka2 Bb8 5. Qd1+ Kf2 (or Kg2) 6. Qc2+ K(any)3, and the queen forks the king and bishop on the next move and takes the bishop on the 8th, and the Black king is able to a move to a more favorable spot. (There are many continuations that all lead to the same result.)
For PC=86 or higher, the game is a draw.
Sometimes White has enough material advantage to be able to give up a piece and still win. If PC is high enough, the best approach may be to force a capture of one's own material to reset the counter. We saw this in the earlier position where White sacrifices a queen. "Forcing" can mean there is no legal move but to capture, or, if White has a ply to spare, it can mean that not capturing results in immediate checkmate. Black meanwhile may try to delay capture as long as possible, especially if it would mean a draw, but might capture earlier if doing so ultimately delays the checkmate more.
The first example is a typical scenario arising with two rooks. The rooks have "flanked" the Black king on neighboring files in preparation for a force-sacrifice. White's best move is 1. Rd3+!, offering the rook, which Black should decline. If 1. ... Ke2, 2. Rf1; if 1. ... Ke4, 2. Rf5. Either way, the Black king is boxed in in zugzwang and must take one rook, resetting PC, and White mates on the 15th move.
In the next example, there is a mate in three for PC=95 or lower: 1. Qg7 Ke8 2. K(any) Kd8 3. Qd7#. If PC=97 or higher, the game is a draw. For PC=96, White can mate in 11 by forcing the capture of the knight: 1. Nd7! Kc8 2. Qb8+ Kxd7, and the king queen vs. king position is a basic mate, in 9. It is in Black's interest to not take the knight immediately.
We earlier gave an example of White having to push a pawn inconveniently early to reset PC. When a pawn is on the 7th rank, this adds a new element, as the pawn will promote when pushed.
In this position, if PC=95 or less, White's fastest mate (in 3) involves herding the Black king to the back rank, then promoting: 1. Kb5 Kb8 2. Kb6 Ka8 3. d8Q# (or d8R#). However, for PC=96 or 97, this takes too long, and White must promote sooner, for mate in four: 1. Kb5 Kb8 2. d8Q+ Ka7 3. Qb6+ (say) Ka8 4. Qb7#. Finally, if PC=98 or 99, White must promote immediately—to a bishop! A queen or rook would stalemate, and a knight would be insufficient material. White then mates on the 25th (!) move.
When one cannot reset by pushing one's own pawn, perhaps because one has no pawns, the next best thing may be forcing the opponent to push a pawn and reset the counter.
The first position is a simple illustration. For PC=97 or less, White can mate in two in several ways, e.g.: 1. Kb3 Kb1 2. Rd1#. For PC=99, it is a draw. If PC=98, White instead plays 1. Kc1!. Now Black's only legal move is to push the pawn. White has gotten the PC reset, but as a cost the pawn is close to promotion, there are several stalemate traps, and White requires careful play just to mate on move 19.
Another, more complicated rook vs. pawn position is provided and elaborated in Kronsteen, which also includes the next position.
This queen vs. almost-promoted pawn is a (tricky) mate in 28 for PC up to 47. Indeed, if PC=47, the main line involves capturing the pawn on the 100th ply, and mating on the next move. For PC=49 and up, it's a draw. But if PC=48, White must switch strategy at move 24. Again, White traps the Black king so that the pawn push (and promotion) is the only legal move on move 26. And Black has a fascinating defense at this point: underpromoting to a knight! This delays the mate 7 more moves (it would otherwise be a mate next move), making the position overall a mate in 34. A more detailed exposition, with full move sequences, is in Kronsteen.
The above examples were mostly told from the stronger player's perspective (White by convention). It is after all nice to be a winner. But Black's counterstrategy is also a big part of the theory. We already saw some reference to that above; e.g., Black exploits White's ticking clock to make a run for the center while White was busy resetting PC. Often, even before White deviates from a main line, Black will make a different move to force a delay of the mate (or even draw if PC is high enough).
The example to the right illustrates some of the subtleties that arise. For PC up to 78, Black should just grab the unprotected bishop, and it's a mate in 13. But for PC=79, Black should run, to any of d5, e5, or e6, which delays the mate one move. Don't worry, Black will still get the bishop; indeed, White must scramble to get Black to capture it.
The real subtlety arises for yet higher PC. For PC=80, the mate can be delayed an additional move (mate in 15), but only with 1. ... Kd5. And for PC=81, the game is a draw, but this time the unique drawing move is 1. ... Ke5! There is no easy explanation of the variation in which king move is best; all I can do is provide lines for both (these moves are not necessarily unique):
|1. ... Kd5 2. Re3 Kd4 3. Re1 Kd5 4. Kb3 Kd4 5. Bd6 Kd5 6. Re3 Kc6 7. Kc4 Kxd6 8. Re1 Kc6 9. Rd1 Kb6 10. Rd6+ Kc7 11. Kc5 Kb7 12. Rd7+ Kc8 13. Kc6 Kb8 14. Rd8+ Ka7 15. Rc8 Ka6 16. Ra8#|| 1. ... Ke5
2. Kb2 Ke4
3. Kc3 Ke5
4. Rb5+ Ke4
5. Bh4 Ke3
6. Rb4 Ke2
7. Bf2 Kf3
8. Kd2 Kg2
9. Ke2 Kh3
10. Kf3 Kh2 draw|
In the PC=80 line, Black did not "have to" take the Bishop on move 7, but that was the best moment, and last chance, to; run, and Black is mated on move 11, the 100th ply.
What follows are a few miscellaneous positions of interest, mostly combining several of the themes above.
This position is not an example of variation in DTM50, just an illustration of how PC can be unexpectedly relevant. One might think that when besides kings there are only pawns, it would not be hard to reset the counter. In fact, under correct Black defense, White cannot capture or push for 20 moves, so this position is drawn if PC is 60 or greater. It is a mate in 37 for any smaller PC.
In the symmetrical two-queen position to the right, White can mate in 4 in numerous ways in unbounded chess. However, if PC has reached 94, this is not enough, and White must sacrifice a queen to preserve the win.
The fastest sacrifice starts with 1. Qa3. Now after any of Black's five legal moves, White can place a queen somewhere where it must be captured. I will leave this as an exercise to the reader! This guarantees a win even if PC=96 in the initial position. In this case Black should answer either 1. ... Kd5 or 1. ... Ke6. Once one queen is taken, it is a mate in 9, so overall this line is a mate in 11.
However, if PC is less than 96, that is, 94 or 95, White does not have to sacrifice so quickly. The extra ply or plies can be used to bring about a more favorable post-sacrifice position.
If PC=94, White instead plays 1. Qf7! Kd4! 2. Qff5! Kc4! 3. Qb4+! Kxb4 on the 100th ply, leaving a mate in 5 (rather than 9) because the remaining queen is much better placed at f5 than a3, and the king is pushed more to the side. Overall, this line is therefore a mate in 8.
If PC=95, there is no time for that, but there are still faster mates. White can play 1. Qa3 as before, and again Black can answer Kd5 or Ke6. If 1. .. Kd5, 2. Qc5+! If Black declines with 2. ... Ke6, 3. Qbf5#, so Black is "forced" to take the queen, and the position of the queen and king is slightly better than the longest line, a mate in 8 instead of a mate in 9, for a total of a mate in 10, an improvement of one. If instead 1. ... Ke6, 2. Qe7+! with the same result.
This won't work with PC=96, since the threat to immediately mate no longer applies, as Black's decline to take is the 100th ply!
There is different mate in 10 line for PC=95: 1. Qe6+. If Black declines and runs, White can mate in 2, again on the 100th ply. So Black takes, leaving again a mate in 9, but it took only one move to get to it.
|PC=93 or less||mate in 4|
|PC=94||mate in 8|
|PC=95||mate in 10|
|PC=96||mate in 11|
|PC=97 or more||draw|
So, sacrifice smart, if you can.
This remarkable position is the most varied in this class. Here is the complete account:
|PC=79 or less||mate in 11|
|PC=80 or 81||mate in 13|
|PC=82, 83, 84, or 85||mate in 14|
|PC=86 or 87||mate in 15|
|PC=88, 89, or 90||mate in 16|
|PC=91||mate in 17|
|PC=92 or 93||mate in 18|
|PC=94||mate in 19|
|PC=95||mate in 20|
|PC=96||mate in 21|
|PC=97, 98, or 99||draw|
There are even surprising internal divisions. For instance, the natural move 1. Rd4 works fine for PC=88 or 89, but loses two moves if PC=90.
Here is a table of the first few optimal moves. Not all are unique, so when there's a choice I try to keep it simple. All lines have diverged by White's fourth move. Moves marked with ≈ have transposed into the line above it.
A king and two knights cannot normally checkmate a king. However, if the latter king has a pawn, there are often mating sequences consisting of immobilizing the king (which would be stalemate without the pawn), and then mating next move. The pawn and the possible move it provides is a liability. Also, in some situations, a pawn might block an escape square for its own king (in fact, one knight might be sufficient to mate if the pawn is on a rook file). However, the checkmate can be not just complicated but long, up to 115 moves in unbounded chess, so many of these "winning" lines will be draws by R50. This is true even if PC=0, and more true for higher PC.
I at first built a "one-sided" tablebase for all configurations of two knights versus a pawn, as my software at the time was not able to build full 5-man tablebases, but the pawn's limited movement allowed me to break the problem into parts. It was one-sided in that it only distinguishes White wins versus non-wins, not Black wins from draws. The goal here is to identify long White wins, and this narrowed scope makes things much easier. I later built a full "two-sided" table with improved software and confirmed these results.
Pawn promotion threatens to require full 5-man tables. However, this too can be broken down if only considering White wins: If Black promotes to a queen, Black can easily draw unless White immediately mates the next move. In a few cases, Black can save the draw by underpromoting to a knight; e.g., it might check White (but see the second position on the left). Promoting to a rook or bishop is pointless as there is no stalemate issue here.
There is a finite list of 2868 positions with a queen on the first rank where White can indeed mate in 1, and among them 322 where Black could avoid immediate mate were the queen instead a knight. I manually looked at all 161 (half by symmetry), and found that in 12 cases White can mate in two, and in the rest it was a draw. (I later confirmed these results by building full tables.) This covers all scenarios where Black promotes. All other "conversions" result in a 4-man position, for which I have already built complete tables.
I then worked backwards rank-by-rank, square-by-square. What follows are my results.
As might be imagined, the effect of PC is felt at lower values than in the examples above. A maximal case is to the right. In unbounded chess, this is a mate in 73. However, in chess this is a win if and only if PC=0, in which case it is a mate in 84.
What this means is that if Black just made a capture, then White can win. But if this position arose after White just captured and Black made a move, then the game is drawn.
Here are possible lines for PC=0 and PC=1, noting when they need to diverge. At move 24, White diverges from unbounded lines, preventing a draw by R50:
|PC=0 or 1||PC=0||PC=1|
|1. Kd5 Kb6 2. Nc4+ Kb5 3. Nc7 Kb4 4. Nb6 Ka5 5. Nca8 Kb4 6. Kd4 Kb5 7. Kc3 Kc5 8. Na4+ Kc6 9. N8b6 Kb5 10. Kd4 Kc6 11. Kc4 Kd6 12. Kb5 Ke5 13. Kc5 Ke4 14. Nc3+ Kd3 15. Nb5 Ke3 16. Kd5 Kf3 17. Ke5 Ke3 18. Nc4+ Kd3 19. Kd5 Kc2 20. Ne5 Kd1 21. Nd3 Kd2 22. Kd4 Ke2 23. Ke4 Kd2 24. Nb4! Ke2 25. Kf4 Kf2 26. Nc3 Ke1 27. Ke3 Kf1 28. Nd3 Kg2 29. Kf4 Kh3 30. Nb5 Kh4 31. Nf2 Kh5 32. Ne4 Kg6 33. Kg4 Kh6 34. Kf5 Kh5 35. Ng3+ Kh4 36. Kf4 Kh3 37. Nf5 Kg2 38. Ke3 Kf1 39. Nh4 Ke1 40. Nc3 Kf1 41. Ng6 Kg2 42. Ne4 Kf1 43. Nf4||43. ... b6 44. Nd3 Kg2 45. Nd6 Kg3 46. Nb5 Kg4 47. Ke4 Kg3 48. Ne5 Kf2 49. Kd3 Ke1 50. Ng4 Kf1 51. Kd2 Kg1 52. Ke2 Kg2 53. Ke3 Kg3 54. Nf2 Kh4 55. Kf5 Kh5 56. Ne4 Kg6 57. Kg4 Kf7 58. Kf5 Ke7 59. Nf6 Kf8 60. Nd5 Kg7 61. Nf4 Kf7 62. Ng6 Kg7 63. Ne5 Kh6 64. Kg4 Kg7 65. Kg5 Kf8 66. Kf6 Ke8 67. Ke6 Kf8 68. Ng6 Kg7 69. Nf4 Kf8 70. Nh5 Ke8 71. Ng7+ Kd8 72. Kd6 Kc8 73. Ne6 Kb7 74. Nec7 Kc8 75. Ke7 Kb8 76. Kd8 Kb7 77. Kd7 Kb8 78. Na6+ Kb7 79. Nb4 Kb8 80. Nd6 Ka7 81. Kc7 Ka8 82. Nc6 b5 83. Nc8 b4 84. Nb6#||43. ... Ke1!
44. Nf2 Kf1
45. N2d3 Kg1!
46. Ke2 Kh2
47. Kf2 Kh1
48. Ne5 Kh2
49. Nf3+ Kh1
50. Ne2 draw|
(50. ... b5/b6 51. Ng3#)
The rook file lines are particularly complicated, so it's perhaps no surprise that most recordholders involve a pawn on the a file. Also, since usually only Black pushing stops the PC (as opposed to a checkmate), the PC values are usually in groups of two. None of these records is unique, although I aimed for typical representatives.
Longest mate: The first position is the deepest checkmate, where Black is mated after 128 moves. This exceeds the record for unbounded chess. In unbounded chess this position is a mate in "only" 85; if PC=0 or 1, it's mate in 108, and for PC=2 or 3, mate in 110. If PC is 6 or more, it is of course a draw. White must make sure not only that the pawn moves before PC reaches 100, but that every "segment" of the mating line between pawn pushes stays under the 100-ply limit. Indeed, on move 40, the position is similar to the one above, in that it's only winnable if PC=0. On move 105, White takes advantage of the fact if Black takes the unprotected knight, Black can be mated in the corner with just the remaining knight.
|1. ... Ka4 2. Ne4 Kb5 3. Na3+ Ka4 4. Kb2 Kb4 5. Ka2 Ka4 6. Nc2 Kb5 7. Nc3+ Kc5 8. Na4+ Kc4 9. Kb2 Kd3 10. Kb3 Kd2 11. Nd4 Kd3 12. Nc6 Ke3 13. Kc4 Ke4 14. Kc5 Kf4 15. Kd5 Kf5 16. Ne7+ Kf4 17. Kd4 Kf3 18. Nd5 Kg4 19. Ke5 Kg3 20. Ne3 Kf3 21. Nf5 Kg4 22. Nd4 Kg5 23. Nf3+ Kg4 24. Ke4 Kh5 25. Kf5 Kh6 26. Ne5 Kg7 27. Kg5 Kf8 28. Kf6 Ke8 29. Ke6 Kf8 30. Ng6+ Kg7 31. Nf4 Kg8 32. Kf6 Kf8 33. Ng6+ Kg8 34. Ne5 Kf8 35. Nc4 Ke8 36. Nc5 Kd8 37. Ke6 Kc7 38. Na3 Kc6 39. Nb3 a4||40. Nd4+ Kc5 41. Ke5 Kb4 42. Ndb5 Kc5 43. Ke4 Kc6 44. Kd4 Kd7 45. Kd5 Ke7 46. Ke5 Kf7 47. Nc7 Kg6 48. Ne6 Kh5 49. Kf5 Kh4 50. Nd4 Kg3 51. Kg5 Kg2 52. Kg4 Kf2 53. Kf4 Kg2 54. Nf5 Kf2 55. Ng3 Kg2 56. Ne4 Kh3 57. Kg5 Kh2 58. Kh4 Kg2 59. Kg4 Kf1 60. Kf3 Ke1 61. Ke3 Kf1 62. Ng3+ Kg2 63. Nf5 Kf1 64. Nh4 Kg1 65. Ke2 Kh2 66. Kf3 Kg1 67. Nf5 Kf1 68. Ng3+ Kg1 69. Ne2+ Kh2 70. Nf4 Kg1 71. Ke2 Kh2 72. Kf2 Kh1 73. Kg3 Kg1 74. Ng2 Kf1 75. Kf3 Kg1 76. Ne3 Kh2 77. Kg4 Kg1 78. Kg3 Kh1 79. Nd5 Kg1 80. Nc3 Kf1 81. Kf3 Kg1 82. Nc4 Kh2 83. Ne3 a3|| 84. Na2 Kg1
85. Kg3 Kh1
86. Kf2 Kh2
87. Ng2 Kh3
88. Kf3 Kh2
89. Nf4 Kg1
90. Ke2 Kh1
91. Kf1 Kh2
92. Kf2 Kh1
93. Kg3 Kg1
94. Ne2+ Kf1
95. Kf3 Ke1
96. Ng3 Kd1
97. Ke4 Kd2
98. Kd4 Kd1
99. Kd3 Ke1
100. Ke3 Kd1
101. Ne4 Ke1
102. Nd2 Kd1
103. Nf3 Kc2
104. Ke2 Kb2
105. Kd3! Kb3
106. Nd2+ Kb2
107. Ne4 Kb3
108. Nec3 Kb2
109. Ne2 Kb3
110. Nec1+ Ka4
111. Kc4 Ka5
112. Kc5 Ka6
113. Kc6 Ka5
114. Nd3 Ka6
115. Ndb4+ Ka7
116. Kc7 Ka8
117. Nd3 Ka7
118. Nc5 Ka8
119. Kb6 Kb8
120. Nb7 Kc8
121. Kc6 Kb8
122. Nd6 Ka7
123. Kb5 Kb8
124. Kb6 Ka8
125. Kc7 Ka7
126. Nb4 a2|
127. Nb5+ Ka8 128. Nd5 a1Q
Longest mate from PC=0: The next position is the deepest mate with PC starting at 0, where it's mate in 112, indeed up through PC=36. For higher PC the mate length could be as much as 117. For PC=43, it is a draw. There are no longer mates with PC=1.
Most mate lengths: This position has twenty-six possible outcomes, depending on PC. White should start with 1. Na5 for PC up to 52, and either 1. Kf5 or 1. N6d4+ for PC 53 through 56.
|PC=0, 1, or 2||mate in 61|
|PC=3 or 4||mate in 63|
|PC=5 or 6||mate in 65|
|PC=7 or 8||mate in 68|
|PC=9 or 10||mate in 69|
|PC=11 or 12||mate in 73|
|PC=13 or 14||mate in 77|
|PC=15 or 16||mate in 80|
|PC=17 or 18||mate in 81|
|PC=19, 20, 21, 22, 23, or 24||mate in 82|
|PC=25 or 26||mate in 83|
|PC=27 or 28||mate in 84|
|PC=29, 30, 31, or 32||mate in 85|
|PC=33 or 34||mate in 87|
|PC=35 or 36||mate in 88|
|PC=37 or 38||mate in 89|
|PC=39 or 40||mate in 90|
|PC=41 or 42||mate in 91|
|PC=43 or 44||mate in 92|
|PC=45 or 46||mate in 97|
|PC=47 or 48||mate in 99|
|PC=49 or 50||mate in 101|
|PC=51 or 52||mate in 103|
|PC=53 or 54||mate in 110|
|PC=55 or 56||mate in 112|
|PC=57 and up||draw|
Highest cost of PC: The position on the right is a mate in 9 for PC up to 90: 1. Na5 Kb8 2. Ke7 Kc8 3. Ke8 Kb8 4. Kd7 Ka8 5. Kc8 a6 (only legal move) 6. Nc6 a5 7. Nc7#. However, if PC=91 or higher, Black's defense draws after 5. Kc8, so White must force the pawn to move earlier. If PC=93, White's attempt fails and Black draws: 1. Nd6 Kb8 2. Ke7 Ka8 3. Kd7 Kb8 4. Nc7 draw. If the game continued, Black would be forced to push next move and White would have mate in 7. For PC=91 or 92, however, this defense being inadequate, Black can instead create a much worse position for White by pushing on move one: 1. ... a5. Now it requires careful play for White to mate—on move 104! The high PC has cost White 95 moves.
Mate in two versus...: A less dramatic version of the above, but easier to follow, is the longest alternative to a mate in two. For PC up to 97, 1. Nc6+ K(any) 2. Nc7#. If PC=99, it is a draw (no mate in one). If PC=98, White forces an immediate push: 1. Nc7! b5, and White will mate on move 13.
I built the first full 5-man tablebases with two knights (confirming the above work), and then moved on to two bishops (only considering cases of them on opposite colors). The most interesting case is when they are against a knight. This was long considered a draw, but tablebases showed the bishops can generally win, at least in unbounded chess. It can require up to 78 moves to mate, including up to 66 reversible moves, so that some of these "wins" are draws in chess. As above, it is almost always White who makes the first reversible move, so the PC values are grouped in two. Here are some records:
Longest mate: The first position is one of hundreds of examples of the deepest mate, in 66. R50 costs three moves; in unbounded chess Black is mated in "only" 63. In a later section, I give a full line for a position where Black is mated in 64 only when PC=0.
Most mate lengths: The next position has 9 possible outcomes. It is almost unique, up to symmetry, in that the only other position with 8 mate lengths is with Black to move and the king instead at c1, in which case Black plays 1. ... Kd1.
|PC up to 69||mate in 19|
|PC=70 or 71||mate in 20|
|PC=72 or 73||mate in 21|
|PC=74 or 75||mate in 22|
|PC=76 or 77||mate in 23|
|PC=78 or 79||mate in 24|
|PC=80 or 81||mate in 25|
|PC=82 or 83||mate in 26|
|PC=84 and up||draw|
Highest cost of PC: The last position has a simple mate in 6 for PC up to 89; the Black knight is too far away to help as the bishops corner the king: 1. Kf5 Kh7 2. Kg5 N(any) 3. Bf5+ Kh8 (or Kg8) 4. Kg6, and Black will be mated in 2 on either g8 or h8. For PC=90 or 91, this mate isn't fast enough, but White can still win by instead chasing down the knight. After 1. Kd3 Na3 2. Bd7! Nb1 3. Bc1 Kg7 4. Kc2 Kf6 5. Kxb1 Ke5, and with the king in the center White's mate takes 18 moves. For PC=92, this defense by Black forces a draw.
Chess grandmaster Pal Benko calls this the "headache ending". It is generally a draw, but the defense is very tricky, and the wins aren't so easy either. In unbounded chess, the mates take up to 65 moves and up to 59 reversible moves, so that some become draws in chess.
Unlike above, it is often (though not usually) the case that a reset comes from a Black capture of the bishop, so the PC values are not always grouped in two. An example of this is on the left. Black is cornered, and for PC up to 93, White can mate in 4 without bothering with the rook: 1. Rc2+ Kb1 2. Rc1+ Ka2 3. Kc2 (any) 4. Ra1#. If PC=97, White must instead capture the rook with the discovered check (1. Rb4+), for a mate in 10. If PC is 98 or higher, the position is drawn. However, if PC=96, a different strategy is available: 1. Kc1 threatens 2. Rc2#, so Black must move the rook to a3 or c1 to threaten the bishop. Then after 2. Rc2+ Bxc3 3. Rxc3 Kb1, PC is reset just in time, and White mates in two, for a total of a mate in 5.
Most positions, however, don't give White much choice in how to proceed, so the strategy is the same as in unbounded chess, and you just hope you don't need more than 50 reversible moves.
Longest mate: The first position is the deepest mate, and a draw for PC>0. It may be played out just as it would be in unbounded chess, and Black is mated in 56. After Black's move, of course, it's a mate in 56 for White for PC=0 or 1.
Most mate lengths: The second position has these possible outcomes:
|PC up to 67||mate in 16|
|PC=68 or 69||mate in 18|
|PC=70, 71, 72, or 73||mate in 19|
|PC=74 or 75||mate in 20|
|PC=76 or 77||mate in 21|
|PC=78 and up||draw|
Highest cost of PC: The next position is a unique record. For PC up to 89, White has a mate in 6 without capturing, starting with 1. Re5!. For PC=90 or 91, White must instead grab the rook: 1. Ra5+! Kb1 2. Bd3+ Kc1 3. Ra1+ Kd2 4. Ra2+ Kxe3 5. Rxg2, and White will mate on move 19—13 moves more. Note that Black is not forced to take the bishop on move 4, and for PC=92 the position is drawn.
If one wants to count it, the last position is one with a much higher cost, for Black. After 1. ... Ra6+, White can "futilely" interpose the rook, and Black mates on move 2. However, we know that if PC is 98 or 99, the interposition is not futile. But in that case Black can instead immediately take the rook, for a difficult rook vs. bishop match-up where Black mates on move 29. Cost: 27 moves.
We can also consider 5-man positions where Black has only a king, while white has a king and three pieces. This might not seem an interesting case, but it is more so when all three are minor pieces. I built tables for these, including the cases of three knights, three bishops, or bishops on the same color, all of which require underpromotion to reach.
The three knights case has the unusual property of there being no variation in mate length due to PC, and indeed the only thing that stops the move count is checkmate. The deepest mate is Black mated in 21.
For other cases, at least one of the pieces can be sacrificed: With two same-color bishops and a knight, the knight is indispensible. With three bishops, the odd-colored bishop out is indispensible. With a bishop and two knights, the bishop is indispensible. With opposite-color bishops and a kinght, any one piece can be sacrificed.
The first position is the deepest mate, for which PC must be 78 or 79. It is the only such position where Black's next move depends on PC: for PC up to 68, Black plays 1. ... Kf3; for PC 70 and up, Black plays 1. ... Kf5!; for PC=69, either move will do. Note that for low PC (up to 62), Black is mated in only 19.
The second position is just trivia; it's a rare case where algebraic notation requires the full starting square. White mates in 14 (of course, for PC up to 73) starting with 1. Nc8d6!.
Finally, we come to a position with the most variation in this family, sixteen different possibilities:
|PC up to 71||mate in 15|
|PC=72 or 73||mate in 16|
|PC=74||mate in 18|
|PC=75 or 76||mate in 20|
|PC=77||mate in 22|
|PC=78||mate in 23|
|PC=79||mate in 24|
|PC=80||mate in 25|
|PC=81 or 82||mate in 30|
|PC=83||mate in 31|
|PC=84, 85, or 86||mate in 32|
|PC=87||mate in 33|
|PC=88||mate in 34|
|PC=89||mate in 35|
|PC=90||mate in 36|
|PC=91 and up||draw|
Chess theorists use a point system for evaluating material, where a piece is "equal to" a number of pawns. The most common system is, bishops and knights worth 3, rooks 5, and queens 9. One interesting question is, where does the king fall on this spectrum?
Of course, the value of the king is infinite, but we can ask reasonably what the power of the king is. For other pieces, power equals value, since that's what material means. But the question of power alone is I believe a meaningful one. E.g., one can ask questions like, how much does it hinder if the king is cut off from the action?
One approach is to add a non-standard ("fairy") chess piece which has the power of a king but no special value, and see how it stacks against other pieces in the endgame. Such a piece is often called a commoner, a guard, a prince, or a man; I will use the latter term, and represent it in diagrams as an upside-down king. I built tablebases to answer this.
Although a king and minor piece (indeed even a king and two knights) cannot checkmate a lone king, a king and man can. The mates take up to 18 moves, with the deepest being Black mated in 18. This suggests the man is stronger than a minor piece.
On the other hand, while a king and rook versus a king and minor piece is generally a draw, the rook has good winning chances against the man. The mates are surprisingly long and intricate, to the point that many are affected by R50. Indeed, some unbounded chess mates require 59 reversible moves; the deepest is mated in 69. The man's drawing chances hinge on forming a fortress in the corner with the man on (e.g.) b2 and the king behind, so the rook can make no progress.
The first position, where the queen icon represents the man, is an example of one of the many deepest mates. It is drawn for any PC>0, and otherwise mated in 63. In unbounded chess, it's mated in 59, so R50 costs four moves. Black should start with 1. ... Mg4!, the unique move that delays the mate, or draws.
The second position has four mate lengths:
|PC up to 85||mate in 10|
|PC=86, 87, 88, or 89||mate in 17|
|PC=90 or 91||mate in 18|
|PC=92 or 93||mate in 19|
|PC=94 and up||draw|
In the first case, White plays 1. Ka3; in the second, 1. Kb4!; in the third and fourth, 1. Re3!. These moves are all unique.
The conclusion therefore is that it's hard to conclude anything.
This is perhaps the ultimate five-man match-up. Careful—and often incomprehensible—play is necessary to escort the pawn to promotion in positions which are winning. The pawn's move forward is slow, and in chess (as opposed to unbounded chess) one must make sure that between each push no more than 49 moves are expended. In unbounded chess, up to 124 moves can be required for mate. In chess, it turns out the need to not trigger R50 results in even deeper mates.
The position to the right encapsulates multiple records:
It is first of all the deepest mate for any PC, one of only three positions where for some PC it is a mate in 138, in this case for PC=96, 97, 98, or 99, where White must immediately push 1. d3 (not 1. d4?, a draw). This yields the deepest mate for PC=0, where Black is mated in 137; in fact, this ensuing position is only a win for PC exactly 0. Here is a line illustrating how White could meticulously escort the pawn to the end:
|1. ... Qe7+ 2. Kg6 Qe6+ 3. Kg5 Qe3+ 4. Kf5 Kg2 5. Qc4 Qf3+ 6. Kg6 Qg3+ 7. Kf7 Qf3+ 8. Ke7 Qb7+ 9. Ke6 Qb6+ 10. Kf5 Qf2+ 11. Qf4 Qc5+ 12. Qe5 Qf8+ 13. Qf6 Qc5+ 14. Ke4 Qc2 15. Qd4 Kf1 16. Kd5 Qa2+ 17. Kd6 Qg8 18. Kc7 Qf7+ 19. Qd7 Qf4+ 20. Qd6 Qf7+ 21. Kc6 Qf3+ 22. Qd5 Qf6+ 23. Kd7 Qg7+ 24. Ke8 Qh8+ 25. Ke7 Qh4+ 26. Kd7 Qg4+ 27. Kd6 Qf4+ 28. Qe5 Qf8+ 29. Kd7 Qf7+ 30. Kc6 Qf3+ 31. Qe4 Qf8 32. Kb7 Qg7+ 33. Kb6 Qf6+ 34. Kb5 Qb2+ 35. Kc4 Qc1+ 36. Kd4 Qb2+ 37. Kd5 Qb5+ 38. Kd6 Qb8+ 39. Ke7 Qc7+ 40. Ke6 Qc8+ 41. Kf7 Qc7+ 42. Kg6 Qg3+ 43. Kf6 Qd6+ 44. Kf5 Qf8+ 45. Kg4 Qg7+ 46. Kf4 Qf6+ 47. Qf5 Qd4+ 48. Kg5 Ke2 49. Qe4+ Qe3+ 50. Kf5 Kd2 51. d4||51. ... Qh3+ 52. Kf6 Qh6+ 53. Ke7 Qg7+ 54. Kd6 Qf6+ 55. Kd7 Qg7+ 56. Kc6 Qf6+ 57. Kb5 Qf1+ 58. Ka5 Qc4 59. Kb6 Qb4+ 60. Kc6 Qc4+ 61. Kd6 Qb4+ 62. Ke6 Qc4+ 63. Kf6 Kc3! 64. Qe3+ Kb4 65. Qe7+ Ka4 66. Qe5 Qc6+ 67. Ke7 Qb7+ 68. Kd8 Qb6+ 69. Kd7 Qa7+ 70. Ke6 Qa8 71. Qd6 Qe4+ 72. Kd7 Qb7+ 73. Kd8 Qa8+ 74. Kc7 Qa7+ 75. Kc6 Qa6+ 76. Kd5 Qb5+ 77. Ke6 Qe8+ 78. Kf6 Qe4 79. Qa6+ Kb4 80. Qb6+ Ka4 81. Qa7+ Kb3 82. Qc5 Qh4+ 83. Ke6 Qe4+ 84. Kd6 Qg6+ 85. Kc7 Qf7+ 86. Kb6 Qf6+ 87. Kb5 Qf1+ 88. Ka5 Qc4 89. Qb5+ Kc3 90. d5|| 90. ... Qc7+
91. Qb6 Qe5
92. Qc5+ Kd3
93. Kb5 Qb8+
94. Qb6 Qe5
95. Qc6 Qb2+
96. Ka5 Qa1+
97. Kb6 Qg1+
98. Qc5 Qg8
99. Ka5 Qg2
100. Ka4 Qg8
101. Kb4 Qg4+
102. Kb5 Qg7
103. Qc4+ Ke3
104. Kb4 Qd7
105. Qc5+ Kf3
106. ... Kg2 107. Qc7 Qg4+ 108. Kb5 Qe2+ 109. Qc4 Qb2+ 110. Kc6 Qf6 111. Qc5 Kh2 112. Kc7 Qf4 113. Qd5 Kg1 114. Kc8 Qc1+ 115. Kd8 Qf4 116. d7
116. ... Qf6+ 117. Kc7 Qf4+ 118. Kb6 Qf6+ 119. Qc6 Qb2+ 120. Kc7 Qg7 121. Kc8 Qg4 122. Kb7 Qg7 123. Qc1+ Kg2 124. Qc7 Qb2+ 125. Kc8 Qa3 126. d8Q
126. ... Qa8+ 127. Kd7 Qd5+ 128. Ke7 Qg5+ 129. Kf7 Qf5+ 130. Kg7 Qg4+ 131. Kh6 Qe6+ 132. Kg5 Qe3+ 133. Qf4 Qc5+ 134. Kg4 Qc3 135. Qd2+ Qxd2 136. Qxd2 Kf1 137. Kg3 Kg1 138. Qg2#
This line is surprisingly unique; although there are many variations, they all transpose back to the main line for quite a while. It is only on White's move 89 that there is a choice of truly distinct lines.
This position also has the highest cost of PC, because in fact for PC up to 93, White has a mate in only 17: 1. Qc6+ Kg1 2. Qg6+, and wherever Black moves, White forces an immediate queen trade, and White can race the pawn to promotion ahead of Black's king. For PC=94, this fails: Black leaves his queen hanging on the 100th ply and draws by R50. Instead, for PC=94 or 95, White plays 2. Qc5+! then pushes to d4 to reset PC; White can mate on move 61 via a difficult promotion. For PC=96, it is a mate in 138, so the high PC costs 121 moves!
The second position is an arguable improvement, as it is a mate in 138 for PC "only" 88. It transposes into the above line at move 18.
In unbounded chess, both these positions after the push are Black mated in 120.
Here is the most varied position, and its outcomes:
|PC up to 41||mate in 58ac|
|PC=42, 43, 44, 45, 46, or 47||mate in 59a|
|PC=48 or 49||mate in 60ac|
|PC=50, 51, 52, or 53||mate in 62c(50/51) ac(52/53)|
|PC=54 or 55||mate in 63c|
|PC=56 or 57||mate in 66a|
|PC=58 or 59||mate in 68ac|
|PC=60 or 61||mate in 69c|
|PC=62 or 63||mate in 71c|
|PC=64 or 65||mate in 72c|
|PC=66 or 67||mate in 74c|
|PC=68 or 69||mate in 75c|
|PC=70 or 71||mate in 80c|
|PC=72 and up||drawa|
After 1. Qe3! Qb8! 2. Kc4, it is now Black who is first to choose a line based on PC, by playing either 2. ... Qa8 or Qc8+; which is best (or if both are equal) is shown in the above table with a superscript a or c, respectively.
Like the previous match-up, this one has many positions where a long sequence is needed for White to triumph, and many would-be wins are drawn by R50. White (with the queen) can rarely force Black to push the pawn, so Black uses the threat of running down the ply clock to delay the mate, but even so pushing the pawn quickly is often better if no draw is possible.
Length records: The first position is the longest for low PC, mate in 98 for PC up to 23, else a draw. It is, interestingly, unique up to symmetry.
The second is one of several similar positions which are mated in 107 for high enough PC, exceeding the record for unbounded chess. It breaks down thus:
|PC up to 28||mated in 91|
|PC=29, 30, 31, or 32||mated in 93|
|PC=33, 34, 35, or 36||mated in 107|
|PC=37 and up||draw|
Black starts with 1. ... Rc4+, and in fact most other positions of this type are simply the rook being somewhere else on the 4th rank.
Most varied: The next position is one of two that are this variable, both with Black to move. There are fifteen outcomes:
|PC up to 36||mated in 52|
|PC=37 to 44||mated in 53|
|PC=45 or 46||mated in 54|
|PC=47, 48, 49, or 50||mated in 58|
|PC=51 or 52||mated in 68|
|PC=53 or 54||mated in 71|
|PC=55 or 56||mated in 72|
|PC=57 or 58||mated in 73|
|PC=59 or 60||mated in 74|
|PC=61 or 62||mated in 75|
|PC=63 or 64||mated in 76|
|PC=65 or 66||mated in 77|
|PC=67 or 68||mated in 79|
|PC=69 or 70||mated in 82|
|PC=71 and up||draw|
The rook and pawn, of course, also have winning chances, particularly if the pawn is far advanced, but if so the route to promotion (or mate) is short, and if R50 affects the game, it is only because the resultant queen and rook vs. rook position requires a long sequence. In the example deepest mate, in 62, White must immediately promote to win, and R50 delays the resultant queen and rook vs. queen mate by 5 moves.
The next position is one with the most mate lengths:
|PC up to 91||mate in 19|
|PC=92 or 93||mate in 23|
|PC=94 or 95||mate in 24|
|PC=96 or 97||mate in 27|
|PC=98 or 99||mate in 42|
Naturally, for PC=99 White immediately promotes to a queen.
The last position has one less mate length, but is in a sense equally varied due to the draw, and furthermore it is unique (up to symmetry) among such positions in having PC cutoffs of both parities. It breaks down as follows:
|PC up to 92||mate in 20|
|PC=93||mate in 21|
|PC=94||mate in 29|
|PC=95||mate in 30|
|PC=96, 97, 98, or 99||draw|
White should start 1. Rd8 for PC up to 93, else Rc8. White's threat of a rook sacrifice (which is never carried out) gains one move for the even cutoffs.
Since "in principle" a player may promote to any non-king piece, building this table requires having tables with up to 6 men except for the 4 against 1 cases. This is done. In practice, of course, promotion is almost always to a queen. If only one player is able to promote, the extra queen usually yields a quick win. When both players can promote, we switch into queen and pawn versus queen, which happens to provide some of the deepest mates and be affected by R50. And, so, it turns out that two pawns versus one pawn provides the very deepest mates among all 5-man endings, indeed deeper than the record in unbounded chess.
The first position is one of four records (up to symmetry) where Black is mated in 141 for PC up to 98. The other three position are minor variants of this one. The strategy is rather pedestrian, with both players simply racing their pawns to promotion, reaching a position where, as we've seen above can happen, Black is mated in 137 moves.
Because every pawn push resets PC, there is not usually much variation in the mate lengths based on PC, the most being seven distinct mate lengths. The second position is an extreme difference (in size): for PC up to 97, White plays 1. Kd2, and easily mates on move 9; but for PC 98 or 99, White must immediately promote and enter a protracted queen and pawn vs. queen endgame mating only on move 138. This cost of 129 moves is a record for 5 men.
In a 6-man ending where each side has a pawn on the same file and they face each other (not past one another), the pawns are confined to the file they start on and cannot promote, at least until there is a capture, at which point we reach a 5-man position. Thus, this type of ending can be relatively easily built once 5-man tables are completed.
Although I was hoping for longer, in the case where both sides have queens, the longest mate at PC=0 is 127 moves; all (three up to symmetry) such cases are rather boring, involving an immediate pawn capture of a blocked Black pawn on d3. For higher PC, there is a mate in 131, illustrated in the next diagram below; here PC must be 44 or 45 for this longest mate. It is mate in 127 for PC=42 or 43, otherwise mate in 107. There are three other (up to symmetry) positions like this, all basically the same.
One fascinating fact about queen versus queen is that R50 is exactly the right fit for it, in the sense that the longest length to zeroing for an optimal line is 100 plies. In fact, there is a single unique (up to symmetry) position where the optimal line requires exactly 50 moves, shown in the second diagram. (Note of course that for many positions R50 may still affect the queen and pawn versus queen position after a capture.) For PC=0, this is mated in 84, the same as in unbounded chess. For nonzero PC it's a mate in at least 85, or a draw (for PC=11 and up).
Another interesting case is a rook versus a bishop. The rook and pawn versus bishop ending is affected by R50 when the pawn is on a center file, as it can get "stuck" at its fifth rank for more than 50 moves. With facing pawns, it is even more common for a position to be reached where the 50 moves are not enough. The deepest mate at PC=0 is mated in 121; the deepest for high PC is mate in 122 (not shown).
These deep zeroings also leave room for much variation in mate length; here is a record with sixteen outcomes:
|PC up to 29||mate in 63|
|PC=30 or 31||mate in 64|
|PC=32 or 33||mate in 65|
|PC=34, 35, 36, or 37||mate in 66|
|PC=38 or 39||mate in 70|
|PC=40 or 41||mate in 71|
|PC=42, 43, 44, or 45||mate in 72|
|PC=46, 47, 48, or 49||mate in 73|
|PC=50 or 51||mate in 74|
|PC=52 to 57||mate in 75|
|PC=58 or 59||mate in 76|
|PC=60 to 81||mate in 80|
|PC=82 or 83||mate in 81|
|PC=84 or 85||mate in 82|
|PC=86 or 87||mate in 84|
|PC=88 and up||draw|
Most other facing-pawn match-ups aren't terribly interesting, as they're either drawn or there is a fairly quick win for the player in a strong position.
My next effort at a 6-man table is to have all four of White's non-king pieces be identical. This reduces the number of positions by a factor of 24, which brings it within my grasp, albeit in one case just barely. I built tables for queens, rooks, bishops, and knights. The last was particularly challenging and required several new techniques to get barely within memory; the output file was over 15 GiB. I'll give one example.
The position on the left has these outcomes:
|PC up to 81||mate in 10|
|PC=82 or 83||mate in 11|
|PC=84||mate in 12|
|PC=85||mate in 13|
|PC=86||mate in 14|
|PC=87||mate in 15|
|PC=88||mate in 16|
|PC=89||mate in 17|
|PC=90||mate in 18|
|PC=91||mate in 20|
|PC=92||mate in 22|
|PC=93 and up||draw|
Although many tablebases don't bother including five against one configurations, on the grounds that they are "trivial", there is in fact a lot to say here. There are many positions where the fastest mate requires PC be at most in the 60's.
This match-up required an enormous table with several billion positions, which was made possible only by how overwhelmingly drawish it is; as noted, against a lone king two knights is a draw unless there is an immediate mate, so in this position the two knights are a liability to the losing player, either blocking a route of escape or preventing stalemate, and this is not common. 99.959% of legal positions are draws.
Surprisingly (at least to me), there are nonetheless mates as long as 9 for this match-up, and there are even positions where mate lengths vary by PC. The granddaddy of all such positions is shown; it is the unique deepest (up to symmetry). Indeed, all mates in 9 start with the Black pieces exactly on those squares (again, discounting symmetry).
After 1. Nc2+ Ka2 2. Nb4+ Ka1 3. Kb3, White is threatening 4. Nc2#. If PC was initially 91 or lower, moving the knight away from the king's escape square only delays one move: if 3. ... Nd2+, 4. Nxd2 N(any) 5. Nc2#; if 3. ... Na3, 4. Nxa3 N(any) 5. Nc2#; if 3. ... Nc3, 4. Na3 (or Nd2) (any) 5. Nc2#. However, for PC=92 or 93, Black plays 3. ... Nc3!. White must capture the knight (with the king!), and while in the other cases this is part of the main line, here it hurts: 4. Kxc3 5. Kb1! Nd3, and there are a few variations, but the Black king is trapped and the surviving knight is barely too far, and White mates on move 9.
For PC=94 or more, the position is drawn.
This match-up is also drawish, but there are many interesting wins. As we have seen, White has good winning chances if a knight can be won, although many supposed "wins" are in fact draws by R50.
The position on the left is the unique (!) (up to symmetry) deepest mate, in 90, for low PC. In unbounded chess, this is a mate in only 67; White can win a knight quickly after 1. Bd3 (many possible lines) and reach a position winnable in unbounded chess, but a draw under R50. Instead, for chess a much more convoluted path is needed for the win, keeping Black on the ropes but not taking prematurely:
|1. Bd3 Na4 2. Bb4! Ng3 3. Bb5 Nb2 4. Ke5 Nd1 5. Be1 Nh1 6. Kf6 Ne3 7. Kf7! Ng4 8. Kg6 Ne5+ 9. Kf5 Nf3 10. Bc3+ Kg8 11. Bc4+! Kh7 12. Kg4 Nh2+ 13. Kh3 Nf3 14. Kg2 Ng5 15. Bd3+! Kg8 16. Bf6! Nh7 17. Bh4 Nf8 18. Kf3! Kf7! 19. Bc4+ Kg7 20. Bf1! Ng6 21. Be1 Ne7 22. Ke4 Kf6 23. Bh3 Kf7 24. Bg2 Ke6 25. Bf3 Kd6 26. Bxh1||26. ... Nc6 27. Bd2 Kc5 28. Bc1 Nb4 29. Bg2 Nc6 30. Ba3+ Kb5 31. Bd6 Ka6 32. Bh2 Kb6 33. Kd5 Na5 34. Bg3 Nb7 35. Be4 Na5 36. Bf2+ Kc7 37. Ke6 Nb7 38. Ke7 Na5 39. Bg3+ Kb6 40. Kd7 Kc5 41. Bf2+ Kc4 42. Bg6 Nb3 43. Bf7+ Kc3 44. Be1+ Kc2 45. Kd6 Nd2 46. Bg6+ Kd1 47. Bf2 Ke2 48. Bc5 Kf3 49. Ke5 Ke2! 50. Ba7 Nc4+ 51. Kd4 Ne3 52. Bb6 Kf3 53. Be4+ Ke2 54. Bc6 Ng4 55. Bc7 Nf6 56. Ba4 Ng4 57. Be8 Kf2 58. Kd3 Nf6 59. Bc6 Nh7 60. Bd8 Kg3 61. Ke4 Kg4 62. Bd7+ Kh5 63. Kf5 Kh6 64. Be7 Kg7! 65. Be6! Nf8 66. Bf6+ Kh6 67. Bf7 Nd7 68. Bd4! Nf8 69. Be8! Nh7 70. Bb2 Nf8 71. Kf6 Nh7+ 72. Kf7 Kg5 73. Kg7 Kf5 74. Kxh7||74. ... Ke6 75. Kg6 Kd5 76. Kf5 Kd6 77. Ke4 Ke7 78. Bb5 Kd6 79. Ba3+ Ke6 80. Bc4+ Kd7 81. Kd5 Kc7 82. Bc5 Kd7 83. Bd6 Ke8 84. Kc6 Kd8 85. Bf7 Kc8 86. Be7 Kb8 87. Kb6 Kc8 88. Be6+ Kb8 89. Bd6+ Ka8 90. Bd5#|
This line is mostly unique up to (and not including) Black's move 25. Lines starting with 3. Ke5 quickly transpose back, as do those with 21. ... Kf6. If PC is close enough to 43, Black has a few other ways to force White to the same outcome from move 19. After the capture on move 26, White is in a position only won because PC=0; this is true of all four moves Black might make on 25. If PC were 1 in the position after the first capture, Black could draw with 72. ... Ng5+!. There are many possible lines from the capture on; one is shown.
Black's drawing line for PC=44 diverges with 20. ... Kg6!. This is also quite convoluted with many variations (not shown here!).
The next position is the deepest mate for any PC. For PC up to 33, it's a mate in 86; for PC 34 through 39, a mate in 87; PC 40 or 41, mate in 90; and PC 42 or 43, mate in 91. It is unique up to symmetry, except that the bishop on d6 could as well be on e7. White starts with 1. Ba3+!. The position is a mate in only 70 in unbounded chess.
Finally, there are six distinct (by symmetry) positions with eleven
possible mate lengths. I'm bucking the trend by adopting one from
Black's perspective, largely because it is unique in this respect
and three of the five White positions are related to it. The second
position breaks down as follows:
|PC up to 46||mated in 43|
|PC=47 or 48||mated in 44|
|PC=49 or 50||mated in 45|
|PC=51, 52, 53, or 54||mated in 54|
|PC=55 or 56||mated in 57|
|PC=57 to 68||mated in 58|
|PC=69 or 70||mated in 59|
|PC=71 to 76||mated in 62|
|PC=77 or 78||mated in 63|
|PC=79, 80, 81, or 82||mated in 64|
|PC=83 or 84||mated in 70|
|PC=85 and up||draw|
It is surprising to me how deep many of these conversions are; in a few positions the fastest mate requires a maximum PC as low as 31.
Finally, the last position is one of many with a PC cost of 63 (the highest possible). White could mate in 2 with 1. Bd7+ Nc6 2. Nxc6#, but only if PC is at most 97. If PC=98 or 99, White must take, and after 1. Kxb2 Kb5, White will mate on move 65.
In two bishops versus knight, there are a few hundred positions with Black mated in 65 or 66, but hundreds of thousands mated in 64. Hence, it is not surprising that long lines often involve conversion to the latter.
I only built tables for bishops on opposite-colored squares. The other case seemed too silly to bother with, as both players' only hope is a foolish smothered mate.
To win here White must either have a lucky quick mate, or an equally lucky way to collect both Black bishops, as in general two bishops versus one is a draw. In spite of these tight limitations, some interesting situations arise.
The first position is one where PC is "expensive". 1. Be8+ Bd7 2. Bxd7# wins easily for PC up to 97, but for PC=98 or 99, White must immediately play 1. Bxf5. There is still a mate threat, so the Black king runs: 1. ... Kb5 2. Kxa1, and White will mate on move 20. Note that 1. Kxa1?? is a draw.
The next position shows how long a conversion can be delayed to benefit mate length... till move 4. After 1. Be5+ Kb1, White might continue with 2. Bc2+ Kc1 3. Bf4+ Be3 4. Bxe3#. But if PC is (in the initial position) 94, 95, 96, or 97, White must take the hanging bishop with 2. Bxf3. Black's best defense is 2. ... Ba3! guarding the key b2 square and trying to bait the White king into letting the Black king run. White plays the tempo move 3. Be2, and Black answers 3. ... Bb2, and 4. Bxb2 would be stalemate. White instead chases the king away: 4. Bd3+ Kc1 5. Bf4+ Kd1 6. Kxb2 Ke1, and, although the king is confined to the back row, it still takes 9 more moves to mate. For PC 98 or 99, the position is drawn; although White can play 1. Bxf3, after 1. ... Bd4!, the Black bishop holds the main diagonal, and White can do nothing.
I first built a table restricting both pairs of bishops to opposite-colored squares (i.e., no promotion necessary). I later built one with Black's on the same color to see if I got any interesting positions. Other than a few where the PC cost was 19 moves, it didn't provide anything notably new.
For this ending, I again don't bother with the case of same-color-squared bishops.
The deepest mate both for low PC and for any PC is shown to the right. This position is unique (up to symmetry), except that the white-squared White bishop can equally be any vacant space on its main diagonal, other than checking from d5. It is a mate in 73 (as in unbounded chess) for PC up to 57, and a mate in 75 for PC=58 or 59.
Two of Black's pieces are largely neutralized, the bishop trapped and the knight pinned, although of course the pin is useless if the position would be drawn by R50 after White takes the knight. Hence, careful, and cryptic, play is needed. For instance, in unbounded chess, White has multiple winning choices for the first 3 moves; in chess, all but one draw in every case. The first moves are 1. Bd5+! Ka3 2. Kd4! Kb2 3. Bg7! Kc1 (or Kc2) 4. Be5!, and then there are several basically similar lines depending on where Black runs the king. White eventually immobilizes it, forcing the knight to move.
The next position is one of only four with seven mate lengths, all of which are very similar, to wit:
|PC up to 83||mate in 20|
|PC=84 or 85||mate in 21|
|PC=86 or 87||mate in 23|
|PC=88 or 89||mate in 25|
|PC=90 or 91||mate in 26|
|PC=92 or 93||mate in 32|
|PC=94 to 99||mate in 35|
White plays 1. Be4+ for PC up to 93, otherwise (naturally) 1. Bxg8.
Meanwhile, the bishop and knight can checkmate too, but it requires neutralizing both the opponent's bishops. The deepest is a unique (up to symmetry) mate in 40 which works for any PC, as it involves immediate capture, shown to the left. More fun are the four positions with three mate lengths (the maximum posible), all simple variants of one another. The second position is a representative.
After 1. Nc3+, if Black plays 1. ... Kc1, White can mate quickly with 2. Bg5+ Be3 3. Bxe3#. For low PC, Black can stall by going to the corner: 1. ... Ka1 2. Bxf2, and now the Black king is trapped. White can move the king to a3, and the surviving Black bishop can do nothing as all the pieces are on black squares; worse, it's a liability preventing stalemate. White winds the bishop to b2 and mates on move 6... for PC up to 95.
For PC=96 or 97, however 1. ... Kc1! is the right move. White cannot mate fast enough due to the "futile" interposition and must take 2. Bxf2. Black then lets the other bishop be taken to free the king from its trap: 2. ... Kd2 3. Nxb5 Ke2, etc., and White mates on move 31.
For PC=98 or 99, White must immediately play 1. Bxf2. As before, Black must sacrifice the bishop to free the king, but can displace the White king doing so: 1. ... Ba4+! 2. Kxa4 Kc2 3. Ne3+, etc., and White mates on move 33.
Note that the knight can also take the bishop immediately, but this is irrelevant as it draws (and not the case in some variants of this position).
This ending has only one symmetry, that of the two knights, so it required a very large table, but is nevertheless drawish enough to complete.
The effects of R50 are felt in the most common method of winning, which is one of the two knights being captured. In unbounded chess, bishop and knight vs. knight endings can last well over 100 moves, but because of R50, the deepest in chess is only mated in 80. In unbounded chess the deepest mates against two knights involve immediate capture of a knight into such a deep position; in chess, however, we get a more interesting situation, as the 80-move cap means that the deepest mates involve expending several moves to capture a knight.
Such a maximal position is shown; it is unique up to symmetry and the position of the knight, which can be anywhere but d5 as long as it can move to c3 next move, to get to the unique (up to symmetry) mated in 85 position. The main theme here is to position the White knight on d5 so that any non-king move allows a capture, and then to move the White king to b3 to immobilize the Black king, so Black must forfeit a piece due to zugzwang. Black is still trapped after this, but White must release the king to avoid stalemate, resulting in a very long checkmate.
The next position is unique up to symmetry in having these six mate lengths:
|PC up to 83||mate in 24|
|PC=84, 85, 86, or 87||mate in 35|
|PC=88 or 89||mate in 40|
|PC=90 or 91||mate in 46|
|PC=92 or 93||mate in 59|
|PC=94 and up||mate in 63|
Finally, the knights can have interesting variation in wins too, of which the third position is an illustration.
As would be predicted by the usual point system, a queen should be evenly matched against three minor pieces, making this match one hopes drawish, which is more or less true. There is also more opportunity for (forced) sacrifices for the player with three pieces. When all the minor pieces are the same, which can only occur with underpromotion, symmetries are available to keep the table size barely manageable. I will look at three knights in the next section.
The first example is one with three bishops, where one of the extra bishops of a color can be spared. For PC up to 97, 1. Bh6+ Q(any) 2. Bx(Q)# is the familiar situation where Black has only a desperate interposition. However, if PC=99, White must immediately take the queen, and after 1. Bxg3 Kd2, the king escapes from the edge and White can only mate on move 10. For PC exactly 98, however, White has another resource: 1. Ba3+! forces 1. ... Qxa3 2. Kxa3 Kb1, and now White mates on move 6.
The second position is an example of the greatest variation; curiously, after 1. Bg7!, White can usually not take the pinned queen immediately (without drawing):
|PC up to 89||mate in 13|
|PC=90 or 91||mate in 15|
|PC=92 or 93||mate in 17|
|PC=94 or 95||mate in 18|
|PC=96 and up||draw|
The queen also has winning chances, in fact generally better chances. If the queen takes a bishop, it may result in a position itself affected by R50. In the (infinitesimally unlikely) case where all three bishops are on same-colored squares, the conversion to take the first bishop is itself in turn affected by R50. An example is the position to the right: it is drawn, but after 1. Qa3+, a position is reached that would be a win if PC were 0.
The slightly more reasonable case where the bishops are split 2-1 on the colors gives us depth records. The next position is mated in 71 for PC up to 50, mated in 81 for PC=51 or 52, and a draw for PC 53 or higher; Black plays 1. ... Kg8 in each case. The first position to the left is the deepest for PC=0; mate in 79 for PC up to 59, else draw.
Finally, the shown most varied position has these outcomes:
|PC up to 71||mate in 26|
|PC=72 or 73||mate in 27|
|PC=74, 75, 76, or 77||mate in 28|
|PC=78 or 79||mate in 30|
|PC=80 to 85||mate in 32|
|PC=86 or 87||mate in 33|
|PC=88 or 89||mate in 44|
|PC=90, 91, 92, or 93||mate in 45|
|PC=94 or 95||mate in 56|
|PC=96 or 97||mate in 62|
|PC=98 or 99||draw|
Again, one would expect the knights to be evenly matched against a queen, but the queen's winning chances are good enough that I was only able to build a table for the case where the knights win. I also built tables for when the opponent has a rook; while it may seem this is unevenly matched, the usual indispensibility of every knight makes the ending fairly drawish.
Generally one knight can be sacrificed only if it allows an immediate mate. More commonly, the knights simply trap the king before the opposing piece can get in the way, or fork and take it. As an example of this latter case involving a queen, this position has the most variation (which is unique up to symmetry and the position of the forking knight):
|PC up to 91||mate in 5|
|PC=92 or 93||mate in 7|
|PC=94 or 95||mate in 9|
|PC=96 or 97||mate in 11|
|PC=98 or 99||draw|
The next position demonstrates how sacrifice might work, although in this case the force-sacrifice is used at the lower PC. For PC up to 96, after 1. Nc3 (Nd2 also works), Black must play 1. ... Qd1 to cover the mate threat on b3; the game then goes 2. Ncb3+ (or Ndb3) Qxb3 3. Nxb3#. For PC=98 or 99, the game is thus a draw. But for PC=97, White plays 1. Nc3! (Nd2 doesn't work) Qd1 2. Nxd1 Kb1, and White will mate on move 6.
Against a rook, the knights have more flexibility, and there are more varied positions, such as the next one with these outcomes, which is unique up to symmetry in having five mate lengths:
|PC up to 91||mate in 8|
|PC=92 or 93||mate in 10|
|PC=94 or 95||mate in 12|
|PC=96 or 97||mate in 15|
|PC=98 or 99||mate in 16|
Curiously, the rook also has winning chances. Usually the strategy is to pick off two knights quickly by hanging mate threats over Black, and getting into a won rook vs. knight position. The deepest is a mate in 41, and there are positions of this depth which are drawn for PC as low as 88. But perhaps more remarkably, the rook may have several choices based on PC, as shown in the second diagram; I leave the PC cutoffs as an exercise.
We now come to six-man positions involving a pawn.
Of course, to compute positions with a pawn one must first generate tables for where the pawn promotes, so to keep things drawish I was only able to do this for White wins. However, that is where the action is, as many deep mates for White exist in this ending, and indeed much deeper than in unbounded chess.
The first position is the unique (up to the one symmetry) deepest mate for PC=0. Black is mated in 151 for PC up to 20.
The second position is the unique (up to symmetry) deepest for any PC. It is mate in 121 for PC=0 or 1, mate in 159 for PC 2 through 9, else a draw. We get therefore the deeper mate if White has squandered a single move.
The final position is an example of the most varied, twenty-two possible outcomes:
|PC up to 20||mate in 60|
|PC=21, 22, or 23||mate in 61|
|PC=24, 25, 26, or 27||mate in 62|
|PC=28 through 37||mate in 63|
|PC=38 or 39||mate in 64|
|PC=40, 41, 42, or 43||mate in 65|
|PC=44 or 45||mate in 66|
|PC=46 or 47||mate in 67|
|PC=48 or 49||mate in 70|
|PC=50 or 51||mate in 71|
|PC=52 or 53||mate in 72|
|PC=54 or 55||mate in 73|
|PC=56 or 57||mate in 74|
|PC=58 or 59||mate in 75|
|PC=60 or 61||mate in 76|
|PC=62 or 63||mate in 77|
|PC=64||mate in 80|
|PC=65||mate in 81|
|PC=66 or 67||mate in 85|
|PC=68 or 69||mate in 90|
|PC=70 or 71||mate in 91|
|PC 72 and up||draw|
In unbounded chess this is a mate in 60, the same as if PC is not more than 20.
We now consider a case where Black has two pawns. When White has two knights, it becomes particularly interesting, because although each opposing pawn is a threat, at least one is necessary to force a win.
The first two positions are depth records for PC=0 or for any PC. Note the former (which is unique up to symmetry) is slightly less than in unbounded chess (for PC up to 79), while the latter (mate in 135) is deeper, and breaks down as follows:
|PC up to 45||mate in 118|
|PC=46, 47, 48, or 49||mate in 120|
|PC=50, 51, or 52||mate in 121|
|PC=53||mate in 126|
|PC=54 or 55||mate in 135|
|PC 56 and up||draw|
Finally, we again consider this position with the (unique up to symmetry) most number of mate lengths:
|PC up to 12||mate in 45|
|PC=13, 14, or 15||mate in 46|
|PC=16||mate in 47|
|PC=17, 18, 19, 20, or 21||mate in 48|
|PC=22, 23, 24, or 25||mate in 51|
|PC=26 or 27||mate in 52|
|PC=28, 29, 30, 31, 32, or 33||mate in 53|
|PC=34, 35, 36, or 37||mate in 54|
|PC=38 through 61||mate in 55|
|PC=62 or 63||mate in 56|
|PC=64 through 73||mate in 57|
|PC=74||mate in 59|
|PC=75||mate in 60|
|PC=76 or 77||mate in 61|
|PC=78 or 79||mate in 64|
|PC=80 or 81||mate in 65|
|PC=82||mate in 67|
|PC=83||mate in 68|
|PC=84 or 85||mate in 70|
|PC=86 or 87||mate in 72|
|PC=88 through 97||mate in 74|
|PC=98 or 99||mate in 75|
This ending also has very deep and tricky wins, although they are less deep than in unbounded chess. The first position is a depth record, with mated in 126 for up to PC=46. There are actually two different themes that both cap at mated in 126; the shown position is the unique deepest for one of these themes.
The second position is the unique most variant position for White to move; if the Black bishop on d1 is on b3, we get the unique most variant position for Black to move, which starts with Bd1. The final diagram illustrates the unique maximum penalty for White to move, where a PC of 92 or 93 costs 103 moves.
Note that the Black pawn is in all cases a rook pawn on its initial square.
For all the following sections, I used Amazon's AWS service to do computations on virtual machines with large amounts of memory.
R50 eliminates many deep wins in unbounded chess in this ending, and elongates others. The deepest win (for any PC) is the first position on the right, which is mate in 65 for PC=0, otherwise a draw.
The second position is an example of a highest cost due to PC. White can win with 1. Rh7+ (or Rh8+) Qh6 (or Qh4) 2. Rx(Q)#, but if PC=98, this does not suffice, and White must play 1. Rh4+! Qxh4+, and with careful play White mates on move 30. For PC=99, of course, the position is drawn.
There is a unique position with 11 different outcomes depending on PC (not shown).
The deepest win for the queen is 51 moves, which is unaffected by R50. There are positions with five different outcomes depending on PC, four mates and one draw (not shown).
This tablebase came with a new kind of challenge, as a computer crash destroyed approximately 10% of the table, and I used a "reconstruction" technique to rebuild the missing part with lesser time and memory requirements. I again restricted attention to positions where the bishops were on opposite-colored squares.
Starting from PC=0, R50 does not affect any optimal before conversion, but its effects are felt on lines where a bishop is sacrificed, reducing reducing to the rook and bishop versus rook ending noted above. So, in most cases R50 merely eliminates deep wins. The first position is the unique depth record for PC=0, a mate in 61.
However, for higher PC we get deeper wins than in unbounded chess. The second illustrates this, and also a theme sometimes called the "eternal rook", or (per Tim Krabbé) the "rambling rook", a rook which harasses the king but cannot be captured without causing stalemate. In the illustrated next position, it merely delays the mate for PC=49 or 50, although for higher PC it can stall to a draw.
Optimal play is shown both for White to win and Black to delay to a draw, depending on PC. Most moves are unique, with only a few mostly trivial variations: 15b, 18w, 22b on the drawing lines, and 23b onward on the drawing line.
|1. ... Rg6 2. Bh2 Rg3+ 3. Ka4 Ra3+ 4. Kb5 Rb3+! 5. Kc6 Rc3+ 6. Kd7 Rd3+! 7. Ke7! Re3+ 8. Kd8! Re8+ 9. Kd7 Rd8+ 10. Kc6 Rd6+ 11. Kc7! Rd7+ 12. Kb6 Rd6+ 13. Ka5 Ra6+ 14. Kb5 Ra2 15. Bf4 Rb2+ 16. Kc6 Rc2+ 17. Kd7 Rd2+! 18. Ke7||PC started at 50||PC started at 51|
|18. ... Rd4! 19. Be5! Rd5 20. Rb3! Rxe5+ etc., as in unbounded chess||18. ... Re2+! 19. Kf6 Re8! 20. Rc7 Rf8+! 21. Kg5 Rg8+! 22. Kh6! Rf8 23. Be5! Re8 24. Bb7+ Kb8 25. Re7+ Ka7 draw|
This perhaps stands as my ultimate tablebase. A host of new techniques were brought in in service of its creation, in particular handling the promotion logic. The promotion to a knight case required a computation of unexpectedly large (and expensive) magnitude; the bishop case required a new kind of gymnastics. The end result is the deepest mates I've yet to find, perhaps the deepest known in chess.
This is the deepest from PC=0:
|1. ... Kg2 2. Kg7 Rb6 3. Bf4 Rb4 4. Bd6 Rd4 5. Bf8 Rd2 6. Bc4 Kf3 7. Kf7 Ke4 8. Ke6 Kd4 9. Bb3 Kc3 10. Bd5 Rh2 11. Kd6 Rh4 12. Kc5 Kb2 13. Bg7+ Ka3 14. Be5 Ra4 15. Bc3 Rh4 16. Bd2 Rh5 17. Bc1+ Ka4 18. Kc4 Rh4+ 19. Kc3 Rh3+ 20. Kc2 Rh2+ 21. Kd1 Rh3 22. Be6 Rh2 23. Bf4 Rf2 24. Be3 Rf3 25. Kd2 Kb4 26. Bd4 Rg3 27. Bf6 Rg2+ 28. Kd3 Rg3+ 29. Kd4 Rg6 30. Be7+ Ka5 31. Bd5 Rg7 32. Bd6 Rg6 33. Bb8 Kb4 34. Be5 Kb5 35. Bf7 Rg5 36. Bd6 Kc6 37. Bb4 Rg7 38. Be8+ Kb6 39. Kd3 Rg2 40. Bf7 Rg7 41. Be6 Kb5 42. a3||42. ... Ka4
43. Bf8 Rg6
44. Bd5 Kb5
45. Kc2 Rg5
46. Be4 Rh5
47. Bg7 Ka4
48. Bb2 Rc5+
49. Kb1 Kb3
50. Bg6 Rc7
51. Bh5 Rd7
52. Ka1 Re7
53. Bd1+ Kc4
54. Ka2 Re3
55. Bf6 Re1
56. Bf3 Re3
57. Bh5 Rg3
58. Be2+ Kc5
59. Bf1 Rf3
60. Be7+ Kb6
61. Bd8+ Kc5
62. Ba6 Rf8
63. Bg5 Kb6
64. Bc4 Rf3
65. Kb2 Kc5
66. Ba6 Kb6
67. Bc8 Kb5
68. Bd2 Rf2
69. ... Kc5 70. Kc2 Rg2 71. Ba6 Rf2 72. Bb7 Kb6 73. Ba8 Rh2 74. Bd5 Kc5 75. Bb7 Rh7 76. Bc8 Rc7 77. Bf5 Rf7 78. Be4 Kd4 79. Bh1 Kc4 80. Bg2 Rf8 81. Be1 Re8 82. Bf1+ Kc5 83. Bc3 Rf8 84. Bb5 Rb8 85. Bd7 Rb7 86. Bg4 Rf7 87. a5
|87. ... Rf4 88. Bc8 Rf8 89. Bb7 Rf4 90. Kb3 Kb5 91. Bc8 Rf7 92. Bg4 Rf4 93. Bh5 Kc6 94. Bg6 Kc5 95. Be8 Rf3 96. Bd7 Rf4 97. Bd2 Rh4 98. Ka3 Rc4 99. Bh6 Rc3+ 100. Ka4 Rc4+ 101. Kb3 Rb4+ 102. Ka3 Rb7 103. Bf8+ Kd5 104. Bg4 Rf7 105. Bh6 Rh7 106. Be3 Re7 107. Bg1 Rg7 108. Bf3+ Ke6 109. Bh2 Rh7 110. Bg3 Ke7 111. Bg4 Rg7 112. Bh4+ Kd6 113. Bf5 Kc7 114. Ka4 Rg2 115. Bf6 Rd2 116. Be4 Kb8 117. Kb5 Ka7 118. Be7 Re2 119. Bc5+ Kb8 120. Bd5 Rd2 121. Bf7 Rb2+ 122. Kc6 Rb7 123. Bc4 Rc7+ 124. Kb5 Rb7+ 125. Bb6 Rd7 126. a6||126. ... Rd1
127. Bc5 Rb1+
128. Ka4 Ra1+
129. Kb3 Rd1
130. Be2 Rd7
131. Bh5 Ka8
132. Kc3 Kb8
133. Bg6 Ka8
134. Kc4 Kb8
135. Be4 Rg7
136. Kd4 Rd7+
137. Ke5 Rf7
138. Ke6 Rg7
139. Kf6 Rd7
140. Kg6 Kc7
141. ... Rd8 142. Kf7 Rc8 143. Ke7 Rg8 144. Bd4 Rc8 145. Bd5 Rd8 146. Be5+ Kb6 147. Kxd8 Kxa7 148. Kc7 Ka6 149. Bc6 Ka5 150. Bc3+ Ka6 151. Bd7 Ka7 152. Bc8 Ka8 153. Bb7+ Ka7 154. Bd4#
This position is mated in 169 for PC=13 or 14, as follows:
|1. ... Rb8 2. Bf6+ Kh7 3. Bd7 Kg6 4. Bh4 Rb7 5. Bc6 Rc7 6. Be4+ Kf7 7. Bd5+ Ke8 8. Kg2 Kd7 9. Bf2 Kd6 10. Bb3 Rc8 11. Kf3 Kc6 12. Ba4+ Kd5 13. Ke2 Kc4 14. Be1 Rb8 15. Bd2 Rb2 16. Be8 Kd4 17. Bf7 Rb7 18. Bb3 Rg7 19. Kd1 Kd3 20. Bc2+ Kd4 21. Ba4 Ra7 22. Bb3 Kd3 23. Bd5 Kd4 24. Be6 Kd3 25. Be1 Re7 26. Bb3 Re2 27. Ba5 Re5 28. Bb4 Rb5 29. Be1 Rb7 30. Kc1 Re7 31. Bb4 Re2 32. Kb1 Rh2 33. Be6 Rh6 34. Bf7 Rf6 35. Bg8 Rf2 36. a3||36. ... Re2 37. Bd5 Rf2 38. Be6 Kd4 39. Bc8 Kc4 40. Be1 Rh2 41. Bg3 Rg2 42. Be6+ Kc5 43. Bh4 Kb5 44. Bd7+ Kc4 45. Bh3 Rh2 46. Be6+ Kc5 47. Be7+ Kb5 48. Bg5 Rg2 49. Bc1 Re2 50. Bd5 Kc5 51. Bb7 Kb5 52. Bb2 Kc5 53. Ka2 Re7 54. Bc8 Re8 55. Bf5 Rf8 56. Bh3 Rf3 57. Bc8 Rf8 58. Bb7 Rf7 59. Ba6 Rf3 60. Bc1 Kb6 61. Be2 Rg3 62. Bh5 Rh3 63. Be8 Rh8 64. Bd7 Rh7 65. Bf5 Rf7 66. Be4 Re7 67. Bd3 Rg7 68. Be3+ Ka5 69. Bf4 Rf7 70. Bg3 Ka4 71. Bg6 Re7 72. Bh5 Kb5 73. Bf4 Kc5 74. Kb2 Rh7 75. Bd1 Rh1 76. Be3+ Kd6 77. Bc2 Rh3 78. Bf4+ Kd5 79. a4||79. ... Rh4 80. Bb3+ Kc5 81. Be3+ Kb4 82. Bd2+ Kc5 83. Bd1 Rh2 84. Kc2 Rg2 85. Bf3 Rg3 86. Be2 Rg2 87. Ba6|
After White's 87 the position is identical to that after White's 71 in the previous sequence, and White will mate on move 170.
While most winning moves during the pawn phase are unique, there are several equivalent ways to checkmate once the material is traded down.
This is so far the only table I have built for an ending with seven pieces. White has five queens versus Black's lone king, where clearly White has an advantage. The effect of R50 is only felt, unsurprisingly, at high PC. Indeed, this ending may seem somewhat trivial, but, besides having the distinction of being a 7-man ending, still has interesting properties. The first illustrated position is a depth record for both PC=0 and any PC, as well as the largest possible penalty for PC, of two moves.
The positions can be broken down into 37 "outcome classes", 15 for White to move, 20 for Black to move, plus 2 counting Black in stalemate or checkmate. Here is the complete list for White to move, with counts of how many positions are in each class, not counting positions equivalent by symmetry:
|277 195 586||mate in 1|
|16 843 079||mate in 2 for PC up to 97, else draw|
|1 288 296||mate in 2 for PC up to 98, else draw|
|13 813 654||mate in 2 for PC up to 97, in 3 for PC=98, else draw|
|22 410||mate in 2 for PC up to 97, in 4 for PC=98, else draw|
|139||mate in 3 for PC up to 95, else draw|
|89||mate in 3 for PC up to 96, else draw|
|102||mate in 3 for PC up to 97, else draw|
|70||mate in 3 for PC up to 98, else draw|
|566||mate in 3 for PC up to 95, in 4 for PC=96, else draw|
|256||mate in 3 for PC up to 95, in 4 for PC=96 or 97, else draw|
|31||mate in 3 for PC up to 95, in 4 for PC=96 to 98, else draw|
|189||mate in 3 for PC up to 95, in 5 for PC=96, else draw|
|51||mate in 3 for PC up to 96, in 4 for PC=97, else draw|
|2||mate in 3 for PC up to 97, in 4 for PC=98, else draw|
Some of these classes contain remarkably few positions. As an indicator of why, consider what would be needed for a position to be mate in 3 for PC up to 97. This means that there must be a White move where Black has the option of not capturing and being mated on the next move, or of capturing and being mated on the third move.
The final class, consisting of only two positions, means that there is one move for White as just described, and also another move that forces an immediate capture where White then mates on the fourth move. One such position is illustrated; the other is the same except for the White pieces on the fourth rank being shifted one square diagonally, to c5 and a5.
Many of these positions make for challenging chess puzzles.
I have so far been able to build 3-man, 4-man, 5-man, and some 6-man tables, as well as the one 7-man table. General 6-man tables remain outside my reach, due to the memory and, to a lesser extent, the time requirements. Software and hardware improvements may make them more feasible in the future, although fully general 6-man tables would require an "unreasonable" amount of memory with the current software.
Due to a typo, a position had been missing from the "Three bishops versus queen" section for seven years. This has been corrected.