Fundamentally all chess moves are a piece moving from one source to another destination including:
- castling as a king move with a distance greater than 1
- pawn moves to the 8th or 1st rank with the additional datum of a new piece
- en passant is the same as a regular pawn capture, it just requires the victim pawn to have moved two squares previously.
Algebraic notation also has an arbitrary and reasonable amount of extraneous detail despite dropping the source location if it's unambiguous.
For example, the captures (x), check(+) and checkmate(#) symbols are all unnecessary given the previous state of the board is always known. With en passant it's also unnecessary to have a special symbol indicating an en passant capture, and indeed there isn't one.
I was initially hoping to get some insight on e.g. which pairs of squares had the fewest moves for a given piece etc.
That being said, I thoroughly enjoyed the video. It was beautifully illustrated and explained everything clearly.
This may not be quite what you were asking for, but it's close, and has the advantage that I can link it right now. Tom7's Elo World chess video has where pieces start and end up, and their survival rates, as a chart: https://youtu.be/DpXy041BIlA?si=Zdh6Rh6mekatp2-q&t=815
So, depending on your definition either could reasonably qualify. Which you pick as the rarest is simply an arbitrary definition.
You could consider different notions, but run into the issue of defining what is unambiguous. IE You could say e2 to e4 is unambiguous for a given game state but that would imply game state must be included in the definition for of a move. Defining what the minimum game state is for an unambiguous move would be a video of its own.
His definition of a move is one ply of algebraic notation. From a chess programming perspective algebraic notation is just a data format and doesn’t have any greater significance.
In programming terms a move is a data structure that allows you to derive one position from another according to the rules of chess.
In Stockfish a move is a 14 bit number, the first 5 bits are the destination square, the next 5 bits are the origin square, the next 2 bits are the promotion piece, and the last 2 bits are the move type (normal, promotion, en passant or castle)
https://github.com/official-stockfish/Stockfish/blob/master/...
Thats 16 bits. Thank you for sending me down that (small) rabbit hole!
I don’t think anyone would actually agree those are the same move.
What Stockfish considers a move includes the full board state, it simply doesn’t need to pass that information around internally. Thus removing that ambiguity but means there’s a great number of moves that have only occurred once.
Also, it’s 6bit + 6bit + 2bit + 2bit = 16 bits which isn’t an arbitrary number. There’s no need to actually encode that something is a promotion because it can be inferred from the board state, but there’s zero cost to pass an extra bit around so it’s included anyway.
- Re4
- Re4+
- Re4#
EDIT
Ok, this seems to do it:
8/3Q4/4R3/6pp/2R2Pk1/6P1/4R1K1/3B4 w - - 0 1
Annoyingly, Lichess uses the rank or file notation in all cases.
> Neither the appearance nor the absence of either a check or checkmating indicator is used for disambiguation purposes. This means that if two (or more) pieces of the same type can move to the same square the differences in checking status of the moves does not allieviate the need for the standard rank and file disabiguation described above. (Note that a difference in checking status for the above may occur only in the case of a discovered check.)
I'd imagine remarkably foolish moves from board states that only quite sophisticated users would get to would be up there
What could happen versus what does happen are entirely different.
Doing some algebraic permutations computation here would be like claiming a 50,000 letter English document has 27^50,000 possibilities.
I mean no, there's words, and they only go in certain orders and there's all these rules.
Here's another approach: humans are pretty lousy when remembering large amounts of anything so let's say there's in practice, only been 100,000 unique games played over and over. Without the help of a computer or careful tabulation, I'm pretty sure no human would realize it because no human can remember 100,000 unique games.
Anyways, it's worth digging into the data to see what the variation really is. I bet the 90th percentile is embarrassingly small with a long tail that's far shorter then most think
edit: so I actually took 7.7 million games from https://www.ficsgames.org/download.html and did some basic processing on them. These are people ostensibly with ELOs over 2000 which is pretty decent just to see if I'll eat crow on this one.
Going in, I was expecting a uniqueness level to be something like 50-70%. Actual percentage of unique games over 7.7 million? 98.7%.
Alright fine.
Although I could try to do 1 billion games, I expected the distribution to be readily visible around 7 million.
Now as an artifact of the data, I made the games as compact as possible, potentially leading to ambiguity maybe. So a game might look like so
a3a5b3b6Bb2Bb7c4e6Nc3Nf6d4Be7Nf3O-Oe3c5d5exd5Nxd5d6Nxe7+Qxe7Bd3d5O-Odxc4Bxc4Nc6Qe2Rad8Rad1Ne4h3Nd6Ba6Bxa6Qxa6Qb7Qd3f6Qc3Rfe8Rd3Ne4Qc4+Kh8Rxd8Nxd8Rd1Qe7Qd5Nf7b4axb4axb4cxb4Qc6h6Rd7Qf8Qxb6Rd8Rxd8Qxd8Qxb4Qe8Bd4Kg8Qc4Nd6Qd5Kh8Nh4Kh7Qb3Qe4f3Qe6Qd3+f5e4g6exf5Nxf5Nxf5Qxf5Qxf5gxf5Bf2Kg6g4fxg4hxg4h5Kg2hxg4fxg4Ne5Kg3Nf7Kh4Ne5Be3Nxg41/2-1/2
number of characters / unique entries / percentage duplicates
99 7470034 0.039416039621657628
98 7462496 0.040385363441781119
97 7454437 0.041421683508957363
96 7446241 0.042475620631500677
95 7437517 0.043597454142611958
94 7428583 0.044746291899176449
93 7419379 0.045929849399894973
92 7409325 0.047222709798876217
91 7399016 0.048548361067336399
90 7388091 0.049953224789125783
89 7376939 0.051387278814333581
88 7364995 0.052923177422393386
87 7352785 0.054493281408027117
86 7340118 0.056122151775432672
85 7326671 0.057851323625950024
84 7312421 0.059683754567414482
83 7297272 0.061631789397747494
82 7281670 0.063638076243272224
80 7247260 0.068062914748240111
79 7228770 0.07044057426456829
78 7209233 0.072952869233227302
77 7187947 0.075690070989017588
76 7166719 0.07841981442939705
75 7144539 0.081271977115830896
74 7119854 0.084446261873027284
73 7094951 0.087648579608837096
72 7068178 0.091091363720824936
71 7039566 0.094770627867995505
70 7009607 0.098623104961001351
69 6978335 0.10264442288391196
68 6944221 0.10703119826195528
67 6909408 0.11150785919986417
66 6871309 0.1164070722832925
65 6831848 0.12148142718723132
64 6790088 0.12685141428305979
63 6744878 0.13266504255419009
62 6698251 0.13866088518630681
61 6648086 0.14511168505848671
60 6594883 0.15195314634822232
59 6540926 0.15889156573829932
58 6481469 0.16653723917595897
57 6419070 0.17456122923325301
56 6355249 0.18276807660975847
55 6282195 0.19216221064468775
54 6209896 0.20145925798763076
53 6133584 0.21127234360201919
52 6048154 0.22225792783565479
51 5963420 0.23315401228435984
50 5870519 0.24510030469790289
49 5770633 0.25794480975187595
48 5668750 0.27104611232094422
47 5558836 0.28518013439112821
46 5443035 0.30007117547551587
45 5323607 0.31542861845637304
44 5192507 0.33228698311784588
43 5064129 0.34879532132158775
42 4925937 0.36656565792950735
41 4777302 0.38567887708631909
40 4626169 0.40511331817237839
39 4465105 0.42582480288508218
38 4301802 0.44682420429097458
37 4126391 0.46938059333470927
36 3948981 0.49219403707682896
35 3770310 0.5151696348833128
34 3581328 0.5394711411415466
33 3387960 0.5643366503548165
32 3202702 0.58815928132701434
31 3008299 0.61315788289287476
30 2810171 0.63863548833641626
29 2617450 0.66341779875536144
28 2413377 0.68965988152851743
26 2044415 0.73710531205655982
25 1849362 0.76218749819167997
24 1674682 0.78464988674290859
23 1499682 0.80715342462054207
22 1324053 0.82973784664289008
21 1167329 0.84989124361622848
20 1005747 0.87066933880105002
19 862832 0.88904701374837569
18 733254 0.90570966192613567
17 599187 0.9229495579983682
16 489782 0.93701812692123954
15 395141 0.94918816879710877
14 300280 0.9613865008348812
13 233128 0.97002168698093183
12 167915 0.97840753392729818
11 115118 0.98519678700915769
10 78474 0.98990889924908909
9 46588 0.9940091724420389
8 26581 0.99658190548385495
7 14316 0.99815908200996462
6 5390 0.99930689103336889
5 2659 0.99965807481590496
4 408 0.9999475346088339
3 180 0.99997685350389731
2 20 0.99999742816709969
1 9 0.9999988426751949
An ancillary analysis would be to compare it to the possible legal permutations at a character count although this would of course require a board and rule model.
I would expect those percentages to decrease as the length increases and perhaps such a function can give more predictive heft to the actual "language" of chess in practice
Proof: Both black and white could move the left rook pawn as their first move and right rook pawn as the second move.
Now reset the board and do right rook first and left rook second. Same board state, different game string.
In practice unique board states is a strict subset of number of moves but given how far off I was on my first assessment ... I wonder if we're talking about another < 2% hit.
All of this is dependent on an actual engine that can process the notation. There's apparently lots of options for pgn.
I'd also like to develop a heatmap based on statistical analysis. I'd imagine this would not only be way less than equally distributed but there'd be no way to slice the data to make it appear equally distributed
- you play bishop a3×e7 taking my queen
- I reply with bishop a7×c5, taking your bishop en passant, getting my queen back (your bishop got taken before it reached my queen)
- you reply with knight a4×b6, taking my bishop while it’s on route to intercept your bishop that took my queen. You get back your bishop, it does end up on e7, and I do lose my queen again
- I reply, taking your knight while it moves through a5. Your bishop dies again, I get back my queen.
- etc.
For knight’s moves, I think you’d have to either make a hard rule as to what square they move over, or let the player say how they moved on every move. Also, two pieces could be taken in one move (a piece on the target square and the knight that just hopped over it)
Standard chess already has some of this in the rules for castling. There, you aren’t allowed to move your king through a position where it would be attacked by an enemy piece. That’s like saying it can be taken en passant.
From: https://www.chessvariants.org/alphabet.html
Standard Wiki/TVTropes-esque warning applies.
Knights don't move through intermediate spaces. They are unlike every other chess piece in that regard.
Also, the pawn can’t deliver checkmate, can it, if it can be taken en passant? It probably is possible to construct a position where taking en passant would bring the king into check in another way, but in those cases, the en passant move isn’t possible.
I believe I've managed to construct a situation where this is the case. The key is that the pawn that would be able to take en passant is being pinned (e.g. by a queen or rook) with the king directly behind it, such that the pawn cannot perform any captures. Then, you just need to make sure all of the squares adjacent to the king are threatened, and finally actually put the king into check via a pawn advancing two squares.
Technically, the c4 pawn cannot be taken en passant (i.e. this is an illegal move) because it would expose the black king to a different check. But I think this is in the spirit of your question.
Contrived game (which you should be able to import into https://www.chess.com/analysis or https://lichess.org/analysis):
1. e4 d6 2. e5 Kd7 3. h4 Kc6 4. Rh3 d5 5. a4 d4 6. Raa3 Kc5 7. Rhg3 Kb4 8. Rg4 Qe8 9. Rag3 Qd8 10. Rg5 Kxa4 11. Rd3 Kb5 12. Ra3+ Kb6 13. d3 Kb5 14. Be2 Kc5 15. Bg4 Qd5 16. Rb3 Qe4+ 17. Ne2 Qxd3 18. Qxd3 Kd5 19. Rb6 a6 20. Nbc3+ Kc5 21. Na4+ Kd5 22. c4#
Or, if you just want to see the end state: blob:https://www.chess.com/20486c7e-a582-41ef-9f1a-bb6fdea2ca36 (seems to work in Chrome, not in Firefox)
(also: https://lichess.org/editor/rnb2bnr/1pp1pppp/pR6/3kP1R1/N1Pp2...)
https://lichess.org/editor/rnbqkbnr/pp3ppp/8/3pp3/2pPP3/4K3/...
the King is never in check, for the purposes of the game. The piece removed isn't in the destination square.
It's just it's own unique rule, born from a period of transition between pawns only being able to take one step and being able to take two.
And yet other pieces can't kill that pawn in passing. Only another pawn.
We shouldn't try to find too much logic.
Practically, game engines model this with an extra state variable in the tree indicating the capturable square.
I'm about equally impressed with the statistical analysis and the video construction and presentation. I can imagine tackling the former, but not the latter. I did notice a few mistakes in the video presentation though (eg a Bd4# that he presented as Be4#). I imagine at some point he just thought "I've polished this thing enough" and stopped.
You'd probably need 15 orders more magnitude of data to get there, but the definition itself is pretty straight forward.
EDIT: so you could define "rare" moves as the biggest difference of occurrences between state N and state N+1.
Computing the number of permutations is thoroughly unconvincing.
For instance, there's 20 possible first moves and of those only probably 2 are played 95% of the time. You can certainly compute what the rates open is and the rarest response that's actually played or the rarest response to the most common open
This would include not only the board state, but all the moves made up to reaching it.
It has a useful meaning for openings.
But it is so sparsely populated by actual games that "rarest" becomes too easy - many moves have been made only once under this definition.
https://en.m.wikipedia.org/wiki/Perfect_information
It might be interesting, but the previous board state should not be a relevant factor when considering rarest move
It's even more interesting because an unknown IRL Chess.com player named Viih_Sou (since revealed to be Brandon Jacobson[2][3]) used this opening to defeat Daniel Naroditsky on May 2, 2024[4] only to be subsequently banned for violating the Fair Play Policy[5].
[1] https://www.chess.com/analysis/game/live/108840009759?tab=re... [2] https://www.reddit.com/r/chess/comments/1claxsm/its_me_viih_... [3] https://en.wikipedia.org/wiki/Brandon_Jacobson [4] https://www.chess.com/analysis/game/live/108394316331?tab=re... [5] https://www.chess.com/blog/Utkarsho/a-grandmaster-account-ge...
I feel like the question remains unanswered
So while I agree that the notation is pretty arbitrary and puts lots of emphasis on how implicitly a move can be recorded I don't think it is fundamentally worse than any other definition. Yes, personally I would have picked something more directly tied to the game than if the notation requires more disambiguation, but I don't think it really makes the video any less interesting or the result and less useful (probably no use either way).
This muddies the waters I think, these doubly disambiguated moves he lovingly isolated would be recorded differently if a marginally involved piece was pinned.
More significantly, there's no SAN notation for stalemate, or en-passant (or many other things of course) of great chess significance, perhaps more worthy of analysis.
The video was still a great technical achievement, and very entertaining for a chess nerd like myself.
Rarestest move for sure.
There needs to be a minimum bar for the data to be meaningful, e.g. by restricting to players above a certain rating threshold, or considering tournament games only.
Plus, restricting the dataset introduces more biases and ambiguities. What exact ELO should be "good enough" for consideration? Why not a point higher or lower? Should they have accounted for time control too, because people in speed chess play worse and can get into weird situations they otherwise wouldn't have been in?
But yes indeed, the single example game he showed was indeed a result of the winner playing very silly moves and the loser allowing it rather than resigning.
At the time of writing based on total games played at https://database.lichess.org/ one in every 272,369 games are mine :p
The rest is a fun programming problem (which he largely glossed over), and it's clear he put a massive amount of care into the video. Thank you!
I bet this video will have a Streisand-like effect [1] on the indicated capture scenarios.
Dig up an old chess book if you don't believe me. You'll find books that only use the long form.
The point is that there are multiple ways of writing the same move, and none of them are more correct than the other. Basing your definition of a move on the precise text that the computer happened to spit out when the games were retrieved, that's just being lazy.
- [what are the commonest checkmating moves](https://chess.stackexchange.com/questions/44959/what-are-the...)
- [Have any full rank-and-file piece specifications ever been required in a masters level game of chess?](https://chess.stackexchange.com/questions/44961/have-any-ful...)
I think this question here is perhaps the more interesting one though, and it's so well explored.
Brings back good memories too, as 10 years ago my first contribution to lichess was to improve the PGN notation for disambiguation. [1]
[1] https://github.com/lichess-org/scalachess/blame/master/core/...
His site is a trove of information for people liking chess trivia.
Creator of the video uses the standard chess notation, which encodes if a move
- is a capture
- is a check
- is a checkmate
- needs a disambiguation
- and other things
Then it turns out that bishop double disambiguation moves are very rare. They require 2 same color bishop underpromotion for the start. And they doing captures and/or checkmates are even rarer. A lot of them never ever happened on lichess.
That's about the content. Now about the format.
The video features hands expressive hands all along, which makes it stimulating to watch. If you are into presentation, go and check it out.
Props to them and all the other chess creators for bringing more influence and people to the game.
It's a 30 minute video. If you care that much about the topic, enjoy. If you just wondered what the move was... you're welcome.
(I wasn't rounding that far, I was giving what I thought was an accurate number. Accuse me of bad info, not bad math.)
Thanks for the TLDR.
FWIW, I wrote the XBoard PGN parser.