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Subject: Win conditions rss

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David Ploog
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I've mentioned a few times that I have been thinking about how to classify win conditions for abstract games. Here is what I mean; I hope it is interesting to some of you. Everything I want to say it most concisely expressed in the following diagram:



Before I proceed with an explanation, three notes:

1. No classification could ever be final. This is because games are created, and someone can (and will!) come up with a new win condition not in the system, or simply combining stuff.

2. This is a simple version, only addressing games with undifferentiated pieces. It is not hard to extend the system, though.

3. Classification is fun, but not overly relevant. It becomes a natural question when you have long lists (on a website or in a book). I wouldn't be too surprised if something like this came up before, but I've never seen it.


To me, the most important distinction is this:

Is the game won by a condition on the board position? or
Is the game won by a score?.

Scoring games need to have a game-end condition, and then a method that assigns a score to the final position. I made three subcategories (I am sure there could be more -- input is very welcome!), and these are

Captive scoring: all sowing games (Awale, Kalah, Buku).

Chain scoring: at game end (typically: board is full), the score of a player is *only* a function of the sizes of chains of that player's colour. Some examples:
Minimize: size of smallest chain
Catchup, Ecalper: size of the largest chain
Produto: product of the two largest chains
Taiji: sum of the k largest chains
Xodd, Yodd: number of chains
Libra: compare number of chains to bid
Omega: product of all chain sizes
Othello, Attaxx, Flume,...: sum of all chain sizes (=just number of stones)
Symple: number of all stones - penalty P for each chain

Territory scoring: At game end, compare enclosed territories. This is vague, but this category is too small for more formal definitions, in my opinion. It contains: Go, Amazons, Storisende. Other candidates might be Anchor, Mobility, Loops.

Now I talk about win conditions determined by board positions. This is the larger part. It turns out that the conditions all depend on the stones of just one colour (logically speaking, this would not have been necessary). I distinguish: is this a condition on the active player or on the opponent? And is this a condition on a single stone, or some stones, or all stones? The major categories I end up with are these:

All pieces of the active player.

Racing: all stones of the player have to be in a distinguished zone. Examples: Halma, Chinese Checkers. An old game with this is Agon (= Queen's Guard) from the 1780s (which has the peculiarity that the distinguished zone is the same for Black and White!). These games cannot really have capturing (in Agon, pieces can be relocated by custodian capture).

Unifying: all stones of the player form a single chain. Examples: Lines of Action, Ayu, Inertia. No distinguished zones. This is often chalked up under "connection", but I think that's misleading.

Stalemate: the active player has no move. This is a generic win condition I am not particularly interested in. If someone has cool examples, fire away!

Some pieces of the active player.

Linking: there is a chain connecting two distinguished zones (typically opposite borders). This is generally meant by "connection". Very many examples: Hex, Slither, Gonnect,... Two interesting cases: Havannah has several and different win conditions. I'd file it here because I think forks are dominating. And Sid Sackson's cool Network game is a linking game, but with a more general notion of "basic link" (which is not "adjacent" but "line of sight"). Similar for Twixt. I think it's better to stick with the simpler notion above which encompasses the huge majority in this category.

Pattern: a bunch of stones of the player forms a specified pattern. The list of patterns tends to be small. No distinguished zones. Examples: Gomoku, Pente, ... (all the n-in-a-row games); Hexade with three patterns (triangle, line, ring); Manalath with the 22 hex pentominos. Morelli's squares also belong here.

One piece of the active player.

Crossing: one stones reaches the distinguished zone. This zone is often the opponent's home row (Epaminondas, Ordo, Breakthrough, ...) but it can be just one corner (Archimedes, Aboyne). I am not aware of a game where the two players have the same goal zone. The white player in Tablut also has this goal, although just for one piece (the king); the zone is the full border of the board. An early representative is Chivalry by George Parker (1888, differentiated pieces) and its later version Camelot (1930, undifferentiated pieces) which has a goal zone of two squares both of which must be filled (so strictly speaking, does not fall under the definition, but I argue as above for Network).

All pieces of the opponent.

Elimination: Draughts and variants, Fanorona, Focus. This is a tricky win condition to get right, which is why elimination games need longer rules (which is a never an assessment about quality, in my opinion).

Stalemate: This is often a secondary win (rather: loss) condition. In Amazons, it's the only goal, but that game is much more precisely classified as a territory scoring game. (The rules are shorter and more elegant phrased as they are.)

Mate, i.e. capturing a particular piece is a win condition on one opposing piece, but requires pieces of several kinds (e.g. Chess and Tablut). It would be possible to have a game like Atari Go where first capture wins but I am not aware of good examples.


I am definitely missing interesting categories. If you have examples, be sure to tell me! Thanks.
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David Bush
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You could edit your post. Change {imageid=4699896} to {imageid=4699896 original} using square brackets of course.
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Russ Williams
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It seems like the diagram confusingly (to me anyway) conflates 2 separate things (how does the game end? how do you win?)

I.e. "point scoring" is presented as how you win when the game ends... but point scoring can also end a game. E.g. "first player to score 100 points wins", and some games' current scores might not be determinable merely by looking at the current board position, e.g. if you earn points during play by capturing enemy groups, and the points earned are e.g. the square of the number of stones in the group.

And the examples of point scoring which you give are all counting features visible on the board at the end of the game, so all seem to fall under the rubric of "board position", because victory in a game like Go or Catchup also depends on the "board position" (whose position is worth more points?), until I read the fine print that you were using "board position" to include how the game ends.
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Michael Van Biesbrouck
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Perhaps the diagram should be a Venn diagram of ways in which the game ends and how winners are determined.

For example, 'opponent has no legal moves' would overlap drawing (stalemate in in Chess), winning (secondary in Santorini, primary in Jotunheim) and losing (Nomic.

Some games such as Santorini have termination conditions that are a matter of how you achieve a board state. You need to know whose turn and what stage of the turn it is to determine if the game has ended. 'Achieving a goal' would be more appropriate than board state.

Games with repetition rules (Chess) may depend on the entire move history.
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christian freeling
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dpeggie wrote:
1. No classification could ever be final. This is because games are created, and someone can (and will!) come up with a new win condition not in the system, or simply combining stuff.

2. This is a simple version, only addressing games with undifferentiated pieces. It is not hard to extend the system, though.

3. Classification is fun, but not overly relevant. It becomes a natural question when you have long lists (on a website or in a book). I wouldn't be too surprised if something like this came up before, but I've never seen it.
It's a good thing that you mention it


Obviously there's some room for improvement and some 'difficult points' have already been mentioned. But it seems a useful way of classifying once those have been ironed out. Discussions like this never reach a conclusion that suits everyone, so in the end its your call.
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David Ploog
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twixter wrote:
You could edit your post. Change {imageid=4699896} to {imageid=4699896 original} using square brackets of course.
Many thanks, done! Now the diagram is very big but hey, I've spent some time on it

russ wrote:
It seems like the diagram confusingly (to me anyway) conflates 2 separate things (how does the game end? how do you win?)

I.e. "point scoring" is presented as how you win when the game ends... but point scoring can also end a game. E.g. "first player to score 100 points wins", and some games' current scores might not be determinable merely by looking at the current board position, e.g. if you earn points during play by capturing enemy groups, and the points earned are e.g. the square of the number of stones in the group.
I'd hope that not, but obviously I cannot argue your confusion away. Do you agree that there's a fundamental difference between games with "sudden death" (board position) and "game end + score tally"?

[Edit: I just realised that your point is why in the diagram it says under "board position" victoriously finished by condition on... (rather than "won by condition on...", say) -- the "finished" is supposed to mean that here game-end is implied. Guess I made a bad job at that.]

Perhaps I should've made more clear that the entire thing is totally subjective. I am aware of that, and I tried to hint at this (in the Amazons and Network examples). Let's look at two more examples:

Abalone: The game is won when one player captures six balls of the opponent. This can be seen as a condition on the board position (because you can simply count how many balls are missing) but it can also be seen as taking scores during the game, and the game ends -- and immediately has a winner -- once 6 is reached.
This method of scoring, using a victory point tracker is extremely common in Euro games, but not so much in abstract games, I would think. I had hoped such games don't invalidate my scheme.

Lines of Action: You could argue that at any given turn, there's a clump score ("clump" is LoA lingo for "chain" or "group"), say "3:4", and the game is won once one side reaches 1. However, I argue that this is not a good way to look at LoA, and it's better to treat unification as a binary win condition. In fact, Ralph Betza in his 1970s article "New ways in LOA" explicitly mentions that clump counting is not how you should play the game...

I agree that there could be games where the score is a more complicated function which you cannot compute from (final position + captured pieces) alone. I've never seen a game of this type which would compel me to write about it, but if there are examples, I'd mention them in my text. Like I said, no system could be complete, there will always be edge cases, and I don't mind.

mlvanbie wrote:
For example, 'opponent has no legal moves' would overlap drawing (stalemate in in Chess), winning (secondary in Santorini, primary in Jotunheim) and losing (Nomic).

Some games such as Santorini have termination conditions that are a matter of how you achieve a board state. You need to know whose turn and what stage of the turn it is to determine if the game has ended. 'Achieving a goal' would be more appropriate than board state.
Yes, of course. I mention that winning conditions are often collections (as in Havannah). For Chess, I'd argue that the major condition is "mate" (a condition on one (particular) opposing piece).

I made this scheme because I was unhappy with how games are classified according to win conditions in previous texts, and simply wanted improve on that. As I see it, the important contribution is the threefold distinction score / board?, followed by your stones / my stones?, followed by all / some / one?. In my opinion, this provides a more systematic approach than, for examples, Cameron Browne's definition of "connection game" (which to be honest, I haven't understood until now, and I've read those sections often). Like I said, this system couldn't be complete, and it couldn't be perfect either (always includes subjective choices!), but I only need it for the games in my book, which are of the more simple-minded kind in this regard...

Oh, and I also think that the formal definition of chain scoring game is really good for something. So that was another reason why I made the posting.

Thanks for input!
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Russ Williams
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To elaborate a bit more on my perception of this stuff:

1. There is the question of "how does the game end?" This can happen because of various reasons, e.g. a player attains some concrete goal (checkmate, connected opposite sides, scored 100 points, eliminated 6 enemy pieces, left the opponent a position with no legal move, etc etc), but it could also happen for various other reasons, e.g. both players pass, one player passes, the board is full, no one has a move, current player has no move, a random event ends the game (if we think more generally than just "our type of combinatorial game"), etc etc etc.

---

2. There is the question of "how do you win?" This can happen by a player attaining some concrete goal (which also has the effect of ending the game), or by seeing who is doing better (by some measure) when the game ends because of various other reasons.

The way to determine who is doing better when the game ends is typically some kind of score (an integer which is points, money, etc, or some more generalized notion of score, e.g. distance to a finish line, number of pieces on special locations, hierarchical or tie-breaker kind of system e.g. "number of widgets owned, and in case of a tie, number of cities owned, and in case of a tie number of armies owned" etc etc).

There needs to be some linear order defined (implicitly or explicitly) on the players to see who is "ahead" at that point, i.e. formally there must exist a suitable function f() mapping players to elements in a linearly ordered set, so that for each pair of players (x,y) either f(x) <= f(y) or f(y) <= f(x). (Or if you want to guarantee that ties are impossible, then f(x) < f(y) or f(y) < f(x).)

The linearly ordered set in practice is usually the integers, but not necessarily - in principle it could be a more "interesting" complicated set which is linearly ordered, e.g. the real numbers, or the lexicographical ordering on the set of strings of characters (e.g. whoever has constructed a word which appears later in the dictionary wins), or whatever...
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Luis Bolaños Mures
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Nice stuff.

dpeggie wrote:
All pieces of the active player. [...]

Unifying: all stones of the player form a single chain. Examples: Lines of Action, Ayu, Inertia. No distinguished zones. This is often chalked up under "connection", but I think that's misleading.

Stalemate: the active player has no move. This is a generic win condition I am not particularly interested in. If someone has cool examples, fire away!
Ayu is actually a stalemate game (and wouldn't work with a unifying goal). Another good stalemate game is Bug, of course.
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christian freeling
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dpeggie wrote:
Territory scoring: At game end, compare enclosed territories. This is vague, but this category is too small for more formal definitions, in my opinion. It contains: Go, Amazons, Storisende. Other candidates might be Anchor, Mobility, Loops.
Sygo is another significant game in this category.
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How would you classify Ren or Olix?
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David Ploog
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BozoDel wrote:
How would you classify Ren or Olix?
Games like these don't occur in my diagram! If forced to, I'd call these "pattern scoring games". For me, it is interesting that they need score tracks -- these are *not* decided by game end condition + subsequent scoring; the scoring has to be done during gameplay. (I guess Catchup is also like this.)

The OLIX rules are neat, I can imagine that game is fun: having several competing patterns and scores for each group certainly spices up the old "first to achieve pattern" concept.

Ren seems too convoluted to me. I wonder how many people have played that. You do know a lot of obscure games. Many thanks

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dpeggie wrote:
BozoDel wrote:
How would you classify Ren or Olix?
Games like these don't occur in my diagram! If forced to, I'd call these "pattern scoring games". For me, it is interesting that they need score tracks -- these are *not* decided by game end condition + subsequent scoring; the scoring has to be done during gameplay. (I guess Catchup is also like this.)
FWIW scoring does not have to be done during play with Catchup. It has a score track only for potential convenience - the score track simply (redundantly) displays your currently largest group on the board.
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David Ploog
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russ wrote:
1. There is the question of "how does the game end?"
2. There is the question of "how do you win?"
Yes, these are two separate questions. In the text, I say something more about this (along the lines of "opening protocol, actual game, game end condition, end protocol (e.g. scoring)". In the section about win conditions, I restrict myself to goals because game-end conditions aren't very exciting for the games that I cover.

However, what I learned from this thread is that I'll add a disclaimer along the lines of "the presentation about games won by the board position assumes that this is also the game-end condition". And I will give some examples where it's not. (And I'll also mention some scoring games where the score is not just a function of the final board position.)

Quote:
There needs to be some linear order defined (implicitly or explicitly) on the players to see who is "ahead" at that point, i.e. formally there must exist a suitable function f() mapping players to elements in a linearly ordered set
Absolutely. However, I will make my text as non-mathy as I can. (For example, in the section about boards, I had already written stuff about "dual graphs", and then I realised I don't need to pester readers with this. If they know about dual graphs, fine. If not, a few pictures will show them how to move from tiles to intersections and vice versa, no theory needed.

Speaking of ordered sets, one bit is interesting, and I do spell that out: the comparison for chain scoring games is often "max", e.g. as in "higher number of stones", but occasionally it is "min", e.g. the smallest chain (i.e. the opposite linear order).

I've lectured about discrete maths, and about topology, but my book on games is *not* the place for that
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And then there are completely different kinds of point scoring, like Pueblo, which are neither capturing, nor territory, nor chains. And points are bad - low score wins.
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dpeggie wrote:
However, what I learned from this thread is that I'll add a disclaimer along the lines of "the presentation about games won by the board position assumes that this is also the game-end condition". And I will give some examples where it's not. (And I'll also mention some scoring games where the score is not just a function of the final board position.)

Score tracks and other components can be considered part of the board state. However, you can't tell whose turn was last by looking at many games, which may determine who has won. Zendo's Spock Rule comes to mind. Perhaps the left side of the diagram could be about move history instead of board state, although winning Santorini requires both.
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mlvanbie wrote:
Score tracks and other components can be considered part of the board state.
Definitely. E.g. besides basic key info like whose turn it is and points earned so far (when that's not deducible from the literal board state alone, e.g. points collected in Hey, That's My Fish! and Through the Desert), many games have significant additional off-board info like your pieces in hand in Shogi; whether a player can still castle in Chess; the pieces you have moved from the central supply to your personal stock in Sploof; money & resources held (e.g. Caylus, Stephenson's Rocket, and other combinatorial euros); etc...

I don't think the literal board state alone suffices to describe the game state of most games.
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Just wanted to chime in and say that I love this line of thinking, and I would love to see (and might contribute to) a wiki page that attempts to exhaustively categorize and cross-link all of these types of end-game victory condition.

I like to use thought-experiments like this one to find combinations that don't already exist.

On the topic of "how does the game end" versus "how do you win", I read the original post (and diagram) as "how do you win", but (I think) each of the branches could also be applied to "how does the game end" instead... resulting (in some cases, of course) in completely different/new games.

Obviously not all permutations will be interesting.

I don't know this definitively (because I haven't done enough reading and/or paid enough attention), but I imagine that something like this categorization was necessary before Yavalath was created.
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David Ploog
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grid wrote:
Just wanted to chime in and say that I love this line of thinking, and I would love to see (and might contribute to) a wiki page that attempts to exhaustively categorize and cross-link all of these types of end-game victory condition.
Thanks! Just a warning: if you try to collect all win conditions, it might end up fractal-like. This is why I went the other way, and condensed goals as much as I could. (Plus the games I am going for tend to be rather simple, so my system is good enough for the book.)

grid wrote:
I like to use thought-experiments like this one to find combinations that don't already exist.
Absolutely! Here is one that I found: in addition to win conditions, you can look at restrictions, particularly turn conditions. By this I mean a restriction you can check after the turn. Examples: diagonal prohibition in Slither; chains of three prohibited in Network; no overlong lines in Yavalath; no 6-omines on Manalath; unification decree for each colour in Ordo (X) and Abande; for both colours in Hive. (You can do all kinds of restrictions but I would guess that turn conditions in this sense are quite clear.) Anyway, while there are various games with unification restriction, there are no games (to my knowledge) with connection restriction! Such a game would need to be a movement game rather than a placement game, but I'm sure it could work.

grid wrote:
On the topic of "how does the game end" versus "how do you win", I read the original post (and diagram) as "how do you win", but (I think) each of the branches could also be applied to "how does the game end" instead...
Yes, this is true. However, I am a lot more interested in actual win conditions because they say very much about a game. I have few examples (again, of games I want to cover) of interesting end conditions. (Buku is one.)
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dpeggie wrote:
unification decree for each colour in Ordo (X) and Abande; for both colours in Hive.
...
Anyway, while there are various games with unification restriction, there are no games (to my knowledge) with connection restriction! Such a game would need to be a movement game rather than a placement game, but I'm sure it could work.
I understand "unification restriction" to mean that all your pieces must be in one connected group each turn like in Ordo. Or are you distinguishing between a "restriction" and a "decree"?

Can you clarify what you mean by a "connection restriction"? Your implicit distinction between "unification" and "connection" is not obvious to me.
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David Ploog
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russ wrote:
Can you clarify what you mean by a "connection restriction"? Your implicit distinction between "unification" and "connection" is not obvious to me.
This is sort of why I wrote all of that

Because I believe that "connection" is an overloaded term, I distinguish between
linking: a chain of the player connects (two or more) distinguished zones; and
unification: all stones of the player form a single chain.

I distinguish these because linking is a condition on some stones and it requires distinguished zones, whereas unification is a condition on all stones which does not require distinguished zones.

As a turn restriction, unification means that all stones (either of each players, or of all stones) form a chain. This is done in Dieter Stein's Ordo, for example. Now linking as a turn condition would mean that after a turn, the distinguished zones are still connected. Imagine a chain of stones marching rubber-like across the board.
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dpeggie wrote:
Because I believe that "connection" is an overloaded term, I distinguish between
linking: a chain of the player connects (two or more) distinguished zones; and
unification: all stones of the player form a single chain.

I distinguish these because linking is a condition on some stones and it requires distinguished zones, whereas unification is a condition on all stones which does not require distinguished zones.
Aha, good, thanks!

(I agree "connection" seems unfortunately confusingly overloaded these days, although I think I used to see it used to mean "linking" specifically and not unification also.)

The idea of a linking requirement is interesting. Indeed, I also am having trouble thinking of any examples of games like that, although it seems like surely some must exist.
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Alright, I have incorporated feedback from this thread into my text. If you're curious, you can find the nine pages on win conditions here: goals.pdf

Two notes:
1. Obviously, this is not finished. I think it's readable already, though.
2. I do not aim for the most general classification. The categories I introduce are all classical, and all I need for the games in the book.

Many thanks to everyone who chimed in!
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russ wrote:
The idea of a linking requirement is interesting. Indeed, I also am having trouble thinking of any examples of games like that, although it seems like surely some must exist.
Chex and Hive spring to mind. Quite a lot of boardless games have an all-pieces-must-remain-connected rule.
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mrraow wrote:
russ wrote:
The idea of a linking requirement is interesting. Indeed, I also am having trouble thinking of any examples of games like that, although it seems like surely some must exist.
Chex and Hive spring to mind. Quite a lot of boardless games have an all-pieces-must-remain-connected rule.
You are still confused as I was about the terminology. This is what David called a "unification" rule, not a "linking" rule.
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russ wrote:
mrraow wrote:
russ wrote:
The idea of a linking requirement is interesting. Indeed, I also am having trouble thinking of any examples of games like that, although it seems like surely some must exist.
Chex and Hive spring to mind. Quite a lot of boardless games have an all-pieces-must-remain-connected rule.
You are still confused as I was about the terminology. This is what David called a "unification" rule, not a "linking" rule.
Ah. In that case, I can't think of any either.
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