Chess Piece Essay: The Royal Question

One of the key features of chess is the royal piece, a piece whose removal means a loss and therefore cannot be left at risk of capture. For orthodox chess this piece is the king, and chess ultimately is a game whose goal is to remove the opponent's king. In practice this finishing blow need not happen; just subjecting the king to the threat of capture with no legal reprieve is death enough. 

For this essay the term royal piece will only be used regarding pieces whose capture signifies a loss in the section where it doesn't move the same way as the king. Until then king and royal piece are interchangeable.

This is checkmate, a logical step to take from how king capture is a loss. Also stemming from this rule is check, or in this case the idea that if the king is under threat of capture removing this threat becomes immediate priority(any move made not to this end is considered illegal and does not stand). Now these seem obvious, but some games do away with these formalities and make king capture the objective, disregarding these extra notions; though leaving your king vulnerable is still a dumb move there is no longer any obligation to do anything about it. This rule also affects another thing which I will now talk about.

If you cannot leave your king vulnerable to capture it also follows that you cannot move your king into immediate capture, rendering them illegal too. There can be situations where no legal moves can be played but is not in check and so has not lost. This is called stalemate and in variant discussions is still a legal quagmire. In orthodox chess it's considered a draw, but this has not always been the case. Stalemate used to be a loss on either end: you lose because you missed out on a chance to mate or he loses because king capture is inevitable nonetheless. The latter shows up in variants that do not invoke check and instead puts capturing the king as the goal.

Is it check?

Lawyer up, boys; there's more.

Consider the pin. If a piece is between its king and an enemy rider it cannot move out of the way as it leaves the king to capture but it can still give check. Pin Chess is motivated by the consideration that a pinned piece can't give check on the basis of such a capturing move being illegal. But the other king goes down first anyway, you'd argue, and this may be true, but this caveat over pinned pieces can only come from questions regarding check; if checkmate weren't an issue the other king will go down first and that will be the end of it.

This brings up something: Can a piece that can potentially move to capture the king but cannot actually check?  The common view is that they do, citing the above instance of priority or the like. But you can also argue that they don't by citing the part about not being able to move anyway. 

An example is pawn promotion. Some variants only allow promotion to a piece that had been captured before, most of the time a pawn that's stuck near the edge of glory waiting for someone to take its spot can check, though it can be argued that if a pawn can't promote it's forbidden for it to cross to the edge of the board and get its upgrade and therefore cannot just go over and sic himself onto the king. This situation only pops up in variants with a finite pool of pieces to promote to, in practice captured pieces.

These only come up in situations that ask "What are the ramifications of not allowing this specific move on the basis that it leads to self-check?" The more common question is the one before it: "Let's mess with checking, why not?"

So while casting and en passant may be rules that variant makers get fussy over, it's nothing compared to the legal quagmire that is check rules.

Anyone can be king

The king's characteristics as a piece are so intertwined to each other that any piece that moves the same way in a variant risks the assumption that it is also royal. "royal piece" and "piece that moves like a king (Betza FW)" are independent qualities, to state the obvious, so we'll fiddle a bit with the latter and see what we can find.

The results aren't much in most cases. In cases of leapers and/or short hoppers the piece stops where it ought to and you simply get a king that moves funny. A change in leap movement does mean different mating ideas, but this doesn't make the target any more slippery; common usage for such pieces is to demonstrate mostly geometric themes. In practical variants there's rarely a need to improve on the king, who is nimble enough on his own.

Speaking of nimble, what of giving royalty to the sliding guys? A royal rider does seem to be a harder one to catch since it will go far away. The common answer is by restricting them from riding past check, i.e. a rider cannot only land on a square where it would be in check but also cannot move past it.

The fact that most variants never bother changing how a king moves but still manages to make playable games probably has got to do with the consequences of having "capture this piece to win" be the main goal to a game.

Pretenders

Royalty is the property of a piece where its capture means a loss, this is the first link in the chain that leads to check rules and the like.

We can change the part about capture to be any other condition a piece has to be in to merit a loss and the substance of royalty still stands, i.e. losing the game by its capture is accidental.

One flip is making checkmate the goal, the premise of selfmate problems. Even as a problem it's hard to play, thus the creation of reflexmate. Orthodox rules on checking still stand, just that what's usually the loss condition wins.

Another way to invert royalty is to make it so that it's in check when not threatened by capture, therefore checkmating it by depriving it of being targeted, a piece you rein in by giving more room.

Parton created another way: a piece that you cannot check but you can put into check which must then be removed by your opponent. If you cannot avoid checking you lose. Since it's easier not to check than check winning is mostly through forces.

Too many kings

The base material of this section is G. P. Jelliss' notes on royal pieces on Variant Chess magazine(VC 4, pp. 37-39). In the last part he mentions situations of multiple royal pieces and that "this is a subject for a future article." The article never did come on the magazine itself, but he has defined it on his glossary.

In such multirex situations a practical approach is to allow capturing royal pieces up to the last man where check stipulations kick in. This gets rid of some of the headaches that may arise from practical play, but as the next paragraph shows, these legalities make for great stipulations.

For chess problems, Jelliss gives three general stipulations. The simplest is a groupmate: The rule that you cannot leave a royal piece in check stands even with multiple of them, so a fork or skewer on two of them is checkmate enough. Groupmate-type loss conditions are also used in practical play.

Supermate is groupmate with the stricter requirement that all royal pieces be checked. If a move checks some but not all the royals is it illegal? The questions asks what happens after this check: If there is no way to resolve the checks then the receiving end has no legal moves left but is not in a loss condition and is considered stalemate. Checking in this way may also be made illegal, but in my opinion is an ad hoc rule.

Monomate only needs you to checkmate one royal, a loose but open-ended idea: how expendable are the others in this case? Must you only exact mate on one royal? Can you check the others? Can another royal stop mate? 

Some games worth mentioning using this multiple royalty madness: Kinglet chess and Extinction chess. Both games require that the every single royal piece be captured, in Kinglet it's pawns while in Extinction at least one piece type must go(e.g. king, queen, both rooks).

Pawns are considered royalty in Kinglet that even promoting a pawn counts as losing it, the question of "are promoted pawns still pawns" only matters a lot in games with drops.

Extinction chess is interesting in that technically every piece is royal, but not equally. On the board are six different multirexes that mingle with one another, but operate in groupmate terms wherever applicable, e.g. forking two rooks is fine but a lone rook is a liability.

But these are nothing compared to All-mate Chess, where capture is through checkmate. To clarify, if a piece is attacked and cannot escaped it through regular chess means, it gets captured in All-mate. This is more a capture variant than a real multirex as king capture is still the goal and multiple captures are settled one at a time.

After rambling on about chess variant pieces I should also start talking more about variants in general. The internet has not only made chess more widespread, but it also gave a new avenue for variants that were only known in small circles mostly through the Chess Variant Pages. Recently, Chess.com and Lichess included variants in the games you can play in real time online, thus there's now more opportunities to play and analyze these hitherto rarely played games.

So, chess variants. Why? Games don't really just pop up out of nowhere (we're not going to get into the whole intellectual property thing) and most of them simply develop over time, where changes come and go until something sticks and we're stuck with it in an official level. The earliest roots of the chess as we know it came from India and its direct ancestor from Persia. Those games also count as variants, and so do the other descendants as they appear in places like China, Japan and Southeast Asia.

In terms of game design no one is really trying to make Chess 2. The classic game as we know it does have its share of problems that designers just can't help but want to tweak, but save for Fischer Random none of the proposals have gotten that much traction. A chess variant is expected to be "chess-like," whatever that means: usually "has king to mate" and "is still abstract strategy" is enough but even those two get done away with when needed.

Variants are also made for experimentation. Ralph Betza has done a lot of work on piece values and has created various games based on a single concept used to its full extent such as variants where pieces move differently depending on some condition it is in. As for compositions, variants come up to make some clever theme a reality, with some variants only working on a problem but not practical play (we'll gloss over retro for sanity's sake). The fun in fairy chess problems is in the use of ideas not possible in orthodox problems, may it be a theme or a geometric trick (some stipulations not in practical play are considered orthodox options in problem circles anyway)

For designers there is a certain paradigm in how a variant should work, while problemists are more interested in conjuring cool lines, whether or not the variant works in practical play. Grasshoppers are a favorite in compositions yet never featured in a dedicated game, but while game designers can't help but love the Xiangqi cannon, they rarely appear in compositions.

More importantly, though, the game has to be fun, or something worth playing in addition to chess. Variants appeal to those who already know chess, and the way they are presented includes allusions to the game everyone knows. There is no sense in trying to pass off a game that feels like chess as some new thing so the best it can do is be a nice extra game for the chess enthusiasts or be another game in the collection for the more general board game folk.

The Chess Variant Pages has already answered the question why play these variants, so I won't add to it any more, but let me conclude this mess this way: Amid the kinds of popular designer board games floating around in the scene, chess variant fans are happy enough to get something new.

Chess Piece Essay: Mix and Match

In the world of generalized chess, anything goes. Chess variants have always introduced new pieces wherever applicable, and if they befit the game's needs. Not every piece will be as one-track as the pieces I've shown earlier; there's no restriction against going crazy with your pieces.

We will deal with hybrids, pieces made from combining movements from leapers, riders and hoppers. To clarify, if a piece does one leap forward and another kind of leap back, it is simply a compound leaper, for a piece to be a hybrid it has to be able to move in two different movement types.

The Simple Mix

Take one movement type and a different movement type, put them together, and you have a simple hybrid. The most notable examples are the knight hybrids commonly explored in more well-known chess variants. The common names for these are the Chancellor (Rook + Knight), Archbishop (Rook + Bishop) and the Amazon (Knight + Queen). As a piece with a Knight and Queen move can be a real monster, some games either try to neuter the piece or place it on a larger board, but there are also games that let the anarchy of an Amazon dictate play, fun but horribly unbalanced.

Not much can be said here that I haven't spitballed on the previous essays; hybrids function as two pieces at once though no one splits hairs at this level.

One Way, then Another

In orthodox chess a piece moves the same way wherever it is on the board whether it is capturing or not, the only piece not to do this is the pawn. 

Pieces whose moves depend on where it is on the board is as much of a topic under pieces as much as it is under board geography. Chinese chess for example has two features that affect how pieces play and some variants use board features to allow some moves that are not possible otherwise, so in this regard the two are intertwined.

A piece moving differently to capture isn't unusual: the pawn for example. The key point here is that the piece only moves a certain way to capture and would not do so otherwise. This then gives us the two general additions: 1) the piece can also capture with its non-capture moves but has additional moves when it captures, 2) the piece has different moves for non-capture and capture. 

Next factor is the direction the piece moves in. On a standard square board there are four main directions: left, right, forward and backward. Board geometry allows more directions than these but in practice these four directions define whether a piece can move a certain way, for example Betza's menagerie is filled with pieces defined this way. Tony Paletta made variants where pieces move depending on destination, and a piece can change movement depending on where it stands (commonly through squares of a color on a checkerboard).

In the hopper essay I have talked about conditional hoppers, though the condition in that case is in whether or not a hurdle is involved. Upon further review conditional hopping is more likely to make a piece a hybrid. In a broad sense this section may be about conditional movement, but convention has it that movement types depending on the piece's state are independent of each other and so we need not get too hung up on classifying such pieces; it is a better exercise to classify the conditions needed instead.

Morphin' Like Morphy

Pieces are expected to stay the same in chess. For these kinds of pieces let's disregard that for a bit and give them the ability to change. Thus follows the first question, into what? Conventionally a piece an only change to a piece that exists in the game; you cannot introduce a piece that should not exist. Next thing to consider is what triggers the change: a capture? being at a certain part of the board? merely moving? 

Parton created a variant that uses the morph cycle of pawn-knight-bishop-rook-queen-king which may have been the standard cycle for a while(not willing to pin Parton as the source yet).

The other way is for a capturing piece to change into the piece it captured, not to be confused with the capturing piece taking in its properties. In the former a rook capturing a bishop becomes a bishop, in the latter the same situation nets a queen.

In any case, the effects kick in after the change is complete

The Piece Moveth, the Piece Taketh

Pieces usually take as they move, that is their capture moves are the same as their normal moves. Displacement capture is taken for granted in chess that changing it is enough to make it a variant. 

Must a piece move to capture? An early variant with a different capture is Rifle Chess, where pieces capture enemies in their range without having to move, 

The first serious attempt at playing with more than one capture property is Ultima, giving everyone in the back row who isn't the king other means of capture, seven in total. Later variations trying to improve on Ultima added extra moves, like Rococo replacing some captures with three new ones. Having pieces move the same but take differently doesn't play like chess but is still chess-like albeit with a lot of queen lines.

A capture that has seen some use is that of jumping over pieces to capture them, analogous to checkers. There's even a variant dedicated to it, reminiscent of earlier variant Dynamo Chess.

Wait, What About the Pawn?

Most chess piece taxonomies label the pawn as a separate thing and sometimes add a sort of separate class of pieces based on pawns. This classification assumes variants that are meant to be played, but a pawn is still a piece anyway. A pawn is a sniper as the only piece to move different in a capture or non-capture, a short rider on the basis of its two-step first move but ultimately hops. Only pawns can promote but if a pawn is given enough power this can be done away with. 

In the end, a piece is a pawn by virtue of being the first line of defense in the starting position, expendable but nonetheless a vital part of the king's army.

Chess Piece Essay: Other concepts

This one here's for the concepts that aren't covered by the previous essays, which are more concerned with movement.

Take it or Leave it

Capture is taken for granted in chess. Capture by replacement is assumed in so many chess variants that it is to be noted whenever a capture doesn't simply involve moving to an enemy piece and taking it, e.g. en passant, Chinese chess cannon. Variant methods of capture are better covered on its own article but it's worth nothing how Ultima is the game that has opened this hitherto unexplored area in chess variants. While not every capture thought up by Robert Abbott has gotten much attention, the mechanics brought up by this game and other inspirations is still open for exploring.

Ultima itself is a worthy first try at pitting capture dynamics against one another, though it has its flaws. The game itself can be described more as a chess-type game with pieces of different powers. Rococo is a great game derived from this idea, keeping some pieces while replacing some with new ones, leading to more piece variety.

Some games have been made based on a change in capture mechanic. An old variation is Rifle Chess, where captures are done but the capturing pieces do not move. Dynamo chess pieces push and pull other pieces instead of simply displacing them, with removal done by pushing pieces off the board. Jumping chess takes pieces by, indeed, jumping on them like checkers. Each of these games have differing dynamics thanks to this change in capture.

All Hail the Piece, Baby

It's not chess without check or checkmate. Most chess variants have a piece that must be protected at all costs. If a piece cannot be allowed to be captured ay any cost, it is said to be royal. In practice, the royal piece is not actually captured, but if it's threatened to be captured next turn, then the piece and player are in check and must get himself out of check on his turn.

In orthodox chess, the royal piece is the king, and there are rules that center over making sure the king doesn't die, mostly involving making sure a move that captures the king doesn't happen. 

The original goal is of course to capture the king, the concepts of check and checkmate stemming from the importance of this goal slowly making the actual act of capture a formality. This paradigm shift has made consequences in legalities that wouldn't exist if one could just simply take the king and end it there. One of them being checkless chess, a simple stipulation with interesting implications.

More legalities abound with stuff like stalemate. Originally a win based on the goal of capturing the king, in modern chess stalemate is a draw thanks to it being a state of being unable to move legally instead of the king being directly threatened. Stipulations between variants vary but are more of a legal requirement; the area where checkmate and stalemate positions matter more is chess composition.

The orthodox chess king moves as a leaper, rather agile for a piece whose preservation is a must (e.g. king geometries in pawn endgames). A more agile piece is any rider, which has given rise to the idea of not letting agile riders move past a square that would put it in check if stopped to make catching them easier. Speaking of riders, it is still an open question if a pinned piece can make a null move that requires it to move anyway.

Quick notes of interest: the anti-king, who is checked if it is not within capturing range; and the contramatic king can move into checked but cannot be checked (in contramatic chess a player unable to do any move other than a check loses). King variations that would only exist as offshoots of the current concept of royalty.

Multiple royal pieces have their own quagmires: What happens when one is checkmated and the other isn't? If both pieces are checked at the same time does that suffice as a win?

Some chess variants do not have a royalty stipulation, such as in Losing Chess or Mock Chess. In these games, the goal is to lose your pieces or capture all your enemy's pieces respectively. How much this deviates from chess is something worth discussing, though it has led to asking if checkers can be classified as a chess variant.

Extinction Chess has a distinct property of all pieces technically having royalty, as the goal is to eliminate every piece of a type. I'm bringing up this game as a borderline example of how royalty works in chess.

A Bit of Reincarnation

Pieces do not return when captured in most variants, a given in most cases. Returning captured pieces developed in Japan through the drop mechanic, the most common form of return mechanism. Parachuting pieces opens up new tactical ideas and is just a fun idea to play with all around.

Within chess composition circles is another way of returning pieces by returning captured pieces back to set squares. Circe chess has gone beyond its original form in 1968 and tons of Circe in various shapes and sizes abound.

These two mechanisms each deserve a piece on their own so I will close this article by noting that Circe is only here rather loosely, most rulesets still allow full capture and in a way the return of the piece onto the board immediately makes it questionable if the capture is a real capture at all.

Stone Placement (and maybe Movement) Games

(An early piece trying to cover types of games that are played with stones and a grid board. Might be better off done as a live catalog of such games.)

Abstract strategy games pride themselves in their relative simplicity, even if such a game were to be embellished in its packaging, it would still be the same as if it were played with household items on a board drawn on the floor. Aesthetically, these games are packaged in the plainest of designs, a bare bones approach to the game where distinctions stay on the text as much as possible.

Equipment for such games need only be made distinct and assigned its task, whatever it looks like is anyone's game. Chess pieces have their roles carved onto them, colors imply ownership if not attribute(for games that use a common pool). One of the simplest yet is a pile of stones distinguished only by color and stones of a like color are owned by one player.

As with every abstract game, it is important to set the terms a bit. In general abstract parlance a stone is a piece that is played by being placed on the board and possibly moving it afterward. A board must consist of cells where a stone must be placed onto, any of that dilly-dally halfway crap is not allowed. Stones of one color are practically indistinguishable from one another and everything that pertains to a stone's capability applies to every other stone bar geographic rules.

Gameplay must involve stone placement, with or without movement. Any starting stones placed on the board are allowed if they aren't the only ones that will be in play. Some would argue that strictly placing stones should be covered here, but this is for the purposes of putting a game where placement and movement happen somewhere.

The most ubiquitous of this form of game uses a Go board and stones, but we can keep the stones and change the board a bit for each example, but that doesn't mean every game listed will work with any type board, e.g. Square grid games have rules that don't work on a hexagonal grid. The game types listed will also be limited, or I'll be here all day splitting hairs.

From Point A to B, or the other one

One of the easiest goals for an abstract strategy game is forming a line between two sides of the board, the sides need not be parallel as long as the only ways of forming the connections are nontrivial. This genre generally splits into two versions, either you try to connect the sides you own or connect any sides that fit the condition.

Cameron Browne has a book on this that I haven't read so I can only give some observations that may or may not have been dealt with in the book.

The purest form of this game is Hex, where players try to connect their two parallel sides of a hexagonal grid rhombus by placing stones on cells. That's it, but some analysis has been done with this game that one of these days we'll find the perfect strategy.

Games of this nature are always designed to achieve no draws by design. I have spitballed about this here, but the way this no-draw thing is done is through:

1. The board geography.
2. The rules of placement. Some placements are banned or can cause changes.
3. Obliging moves. If passing is rarely an option either player has a chance to shoot himself in the foot.

An interesting subset of connection games has appeared where the design goal is to try to make a connection game work on a square grid. Square grids are notorious for their eight-direction connections that either force paths to intersect each other, or forsaking diagonal connections not allowing any connection at all. 

The shape created by two stones of different colors crossing each other in a checkerboard patter is called a crosscut and any purely orthogonal connections are immediately severed once this shape comes into play

From downright banning crosscuts, adding connection rules, allowing captures and whatever could work, the connection game genre has flowered in tougher landscapes.

Not that other board shapes had not been used, but as with the case for hexagonal grids, there's little need to meddle with it except by having some fun with the rules or using a different layout. Practically, regular tilings are already there ready for use, but connection games are a breeding-ground for funny geometries that other game types don't have a demand for.

Ding Ding Ding Ding!

Much simpler is to extend the premise of the classic game Tic-Tac-Toe and have games where the goal is some stones in a row. Obviously the goal is to line their own stones before the other can.

Gomoku is the simplest of this wider generalization, although this has fallen out of favor as being first in turn is too much of a headstart. Attempts to level the balance are solidified in Renju.

Pente has given the whole n-in-a-row game a new twist with its capture rule, even ensuring that the game doesn't devolve to mindless capturing by practically limiting it as a win condition.

This is My Claim

When it comes to games where stones are placed to demarcate territory there's no arguing over Go, but this doesn't mean that other ways to play territory are no longer up for grabs, just a bit unnecessary given that none of them can ever be a contender.

Unsurprisingly, language of games that revolve around stone placement use the language of Go to explain things, even using similar equipment. We'll stick to them as long as there are no equivocations.

Practically, some of the games that Luis Bolaños Mures have designed use territorial concepts but rely on it being a connection game. Whether such games work in a territorial sense (from most enclosed spaces to most stones placed on the board) is yet to be tested, though the bigger the territory the more likely you can build a bridge on it.

Counting score in Go may take a while to learn, but in essence a shortened form of counting spaces claimed by placing stones on every point. Simplified scores(as befits the game) include counting groups of stones, counting the largest group or even going back to raw stone count. Mixing these criteria does happen and you get wild equations to reach a score.

One can also win these games by being the last player to move, which, depending on the size and gameplay, good luck. 

While this may be considered a voting game, Majorities does have a mechanic where a majority-claimed row/direction counts toward a player's vote count, and this also gives a better demonstration of how small moves affects the whole game.

There will be more games with these items that will be made, new mechanics, geometries, concepts. After all, how can you go wrong with such simple pieces? It's as simple as it can get.

Shogi has given inspiration for a number of chess variants centered on the drop capability. Curiously, I have yet to find compositions for such variations. I may not be looking deep enough, but in this article I intend to propose a way to compose problems with drops.

As mentioned, one of shogi's most well-known feature is the ability to return a captured piece by placing it onto the field of play, this drop move being a move in itself. Putting drops in western Chess has been tried, with Chessgi and Crazyhouse being the main forms taking a direct borrowing of the capture-and-drop aspect, though they cannot readily be played on-the-board. Another well-known variant is Hostage Chess where equipment limitations made use of an exchange mechanism to allow drops without any piece changing color.

Taking cues from Japanese tsumeshogi, this proposal is for a form of directmate problem with conditions as follows:

1. Every move by White must be a check

2. If a variant is named(e.g. Chessgi, Crazyhouse), it is implied Black has possession of every other piece not under White's control. In any other case the composer must make an account of every piece in play.

3. The mating position must leave White without any pieces left to drop.

4. Black cannot simply drop a piece to delay an inevitable mate, i.e. an interfering drop that the mating side can capture with no effect to the mate.

Chess Piece Essay: Hoppers

We now enter a realm almost solely within the world of generalized chess. A class of chess pieces unknown to those familiar only with the usual chess pieces.

Hoppers are the name given to a class of chess pieces that require another piece to move and set the direction of movement. A piece a hopper uses to hop is called a hurdle. This means that a pure hopper cannot move alone on an otherwise empty board.

In this piece I will cover linear hoppers, mainly sorted by its action before and after hopping. Long hoppers ride after their hop, while short hoppers stop after landing on its destination square. Leaper hoppers can only start their hop with a single leap, while rider hoppers can ride before they jump. 

Conditional hoppers require a specific hurdle or require a hurdle to fulfill a different task, the latter form usually used in hybrids. The hurdle can remain unscathed or may change state after a hop, the latter making the hopper a hurdle-changing one.

Some Starting Examples

Since the concept of a hopper is relatively alien to anyone not into chess variants, let's start with the quintessential hopper, the grasshopper. A short directional hopper, it jumps over any piece moving as a queen, landing at the square just after the piece and stopping there.

Chinese cannons are pieces in Xiangqi that take a rook's move but needs to hop on a third piece to capture (thus making this a hybrid). Korean cannon hops are long as they need to hop onto a piece before moving continuously, thus making them akin to long grasshoppers.

For a more general approach, the line a hopper takes need not be constrained by rook and bishop lines. The simplest example is the Equihopper, whose hop consists of a leap over a hurdle and landing the same distance. 

Capturing the Gist of It

Hoppers generally capture the (first) piece beyond the hurdle within its range, but some hopper types might put this into question.

In most cases hoppers takes as they move, whether long or short, through replacement. A long hopper stops on the square where it captures. 

Now if a piece has to land on the square before a hurdle, can it capture? This is more a question of how chess spaces work.

It Takes Three to Tangle

Two-hurdle hoppers are very specialist and used usually to prove a point. The obvious development from this is making a hopper jump over exactly two pieces.

A piece that needs to move between two pieces still counts as a hopper, though how this would work is beyond this essay's capabilities.

Let's Make Weird Things Happen

Hoppers can give rise to unusual situations in generalized chess geometry. The requirement that a hopper have something to hop leads to cases where a piece cannot move to a square as it would be check, what is usually called an anti-pin. Anti-pins also happen when a check cannot be responded by a recapture as the recapturing piece will become a hurdle enabling a hopper to capture the royal piece.

It's assumed that the hurdle doesn't change when it is hopped upon, in order to spare many a headache. Hurdle-changing hoppers affect the hurdle, independent of replacement capture capabilities. How the piece changes can range from a simple transformative cycle to a progressive one (i.e. there's a point the hurdle stops changing). What a piece can change into may even depend on the direction of the hop. This classification also counts capturing hurdles as a hurdle change.

Legality of a hop of this kind depends on whether the change in the hurdle creates a legal position, this is independent of the legality of the hop as a move.

Chess Piece Essay: Leapers and Riders

This is the first part of a possibly continuous series of pieces on, well, chess pieces. Future posts might also deal with other aspects of chess variants or generalized chess, but all these will simply attempt to create a taxonomy and/or set definitions regarding some concepts of generalized chess.

For this first essay on pieces, we will look at a basic kind of piece, the Leaper. A leaper is a piece that moves from one space (its origin) to another space a set distance and direction away (its destination). The move is instantaneous and is not impeded by intervening pieces.

Leap Distances

There are two ways to define the distance of a piece's move: destination coordinates and length of leap. Coordinates measure a (x,y) distance from origin to destination on a lattice grid. The simplest way to put it is to imagine the piece leaping x squares in one direction then y squares perpendicular. Length of leap measures the straight line drawn from the origin to the destination in this lattice grid (On a chessboard the points will be the centers of the squares).

Leapers are either simple or composite based on how many unique coordinates it has.

For example, a (1,1) leaper has a leap distance of square root of 2, while a (1, 2) leaper, a knight, has a leap distance of square root of 5. Compounding coordinates and lengths will give more types of leapers. These distance systems do not take into account direction and therefore cannot define leapers with distance constraints.

A well-known piece named from its leap distance is the root-fifty leaper, whose leap distance of sqrt-50 has two coordinates (5,5) and (1,7)

Leap Directions

While the methods of the previous section help with defining the distance of a leap, pieces can also be restrained by direction. Directions of limited leapers are defined relative to the perspective of the mover.

Shogi variants are peppered with examples of limited hoppers, usually relative to a king's movement, e.g. gold and silver general.

Sometimes this leads to a piece that can only move in one direction and might require some extra provisions exclusive to them lest they become deadweight.

Board Range

A chess knight can visit all 64 squares of a chess board once, but other leapers are limited in where they can go. A way for me to gauge a piece's range is to start with a piece on a random square on the board and color the squares based on the least number of leaps needed to reach a square. The board will either be filled with color or contain untouched spots. 

Whether a piece should have full range or not is within the decision of the designer, but the utility of pieces that cannot traverse the whole board is a topic usually glossed over. For these series, pieces that have full board range are "free-moving", otherwise they are "constrained."

Constraints

The most common piece constraint is that of colorboundedness, the state of a piece that can only traverse one color on a checkered board. For the purposes of this essay any piece more constrained than this is seen as heavily constrained. 

From one color to another

A chess knight can only leap to squares that are not the same as the square it is on, while a king can move to a square of either color. While the consequences of the properties of these moves are sometimes mentioned in chess study (e.g. the knight can never lose tempo), I am yet to see any further talk in regards to chess variants (geometries involved in these sorts of leaps may be nontrivial).

This alternating leaping is different from a colorbound one only in the sense of destination squares, as colorbound leapers practically move on a board of their own and can be subject to the same tempo issues.

The question then, if a leaper can always go to either color square, can it always lose tempo?

The Rider

Riders are pieces that move continuously through unoccupied squares. A rider is blocked by a friendly piece and stops moving when it captures. The rider is constructed as making successive hops, with pieces in their trajectory able to intervene. A rook and a bishop are prime examples of simple riders, the queen a compound rider. 

To make sense of a rider being made of successive hops, let our example be the knightrider, which makes continuous knight hops. Just like an actual knight it hops over pieces not within its ride, i. e. only pieces on squares within a successive line of knight hops will matter in its trajectory.

If a piece can be blocked on its way to a square it can go to by virtue of a piece getting in the way of its path, it's a rider. A leaper can only be blocked by friendly pieces on its destination squares. 

Riders long and short

A long rider can travel to its full extent in any direction it goes, blocked only by friendly pieces, captures and the edge of the board. All orthodox chess riders ride long. A short rider has a finite range, i.e. riders with any limited velocity are short may it be constant or maximum. 

In Chess with Different Armies, one of the armies features a short rook, in this case a rook that can only move up to four squares when it moves. 

A change of trajectory

A rider is bent if it requires that in the middle of its move it has to change direction. Bent riders are rarely long, although notable are riders that can diagonally bounce against walls. 

A short bent rider has a finite ride that cannot be extended and must traverse through all the squares or else it is blocked. The knight in Chinese chess is short and bent in this regard. Long bent riders are possible in practice by requiring every step of the ride to bend a certain way, either making the ride a zigzag or a loop. Other regular patterns can also be done

As a more general extension of the short bent rider concept, let's look at a piece called the Sissa. The sissa moves by first riding a set number of squares as either a bishop or a rook, then changing direction and moving the same number of squares in a non-orthogonal direction. If a piece is in the way of the sissa that cannot be captured, that move is blocked.

To give an example on the necessity of limiting long bent riders, consider a piece that can make multiple knight moves in succession, regardless of direction as long as it doesn't eventually land on the same square and it captures at the end of its turn. Considering that a knight can visit an entire chessboard, how different is thus souped-up knight from a piece that is defined by leaping to any other arbitrary square?

10 years of Shibumi (Draft)

In an attempt to conceptualize his studies on distilling game rules and using computers to synthesize them, Cameron Browne designed a fully-defined game system that is compact enough for computers to handle spatially but with enough complexity to be interesting. These efforts gave birth to a 4x4 board with stackable balls: Shibumi.

As a physical set, one only needs a board consisting of sixteen cavities in an evenly-spaced 4x4 arrangement and balls of the same size in three different colors: black, white and red. A possible fourth color, yellow, has been floated as a possible addition. Visually, the set is peak minimalism, even if you play with various balls and boards it still stands on its own, the spheres commanding its aesthetic. Its main feature though is its inherent depth: All of its 30 points are not fully available at once, these points are only available under conditions inherent to the physical set.

Add a precise glossary of terms and a the Shibumi system is ready for play. Keeping things simple, these features are inherent in the physical nature of the set and come intuitively in play. True to its abstract nature, nothing is left to chance and only moves with one certain outcome are legal, though games with randomness have utilized other factors instead.

To give the new system more bang for its buck, a contest was held to collect an initial set of rules made by humans, not only to seed the algorithms but to give the fledgling set a bunch of games for people to play with on the get-go. While Shibumi may be simple enough for someone to rediscover its concepts by scratch, a starting corpus of games is a necessity the same way you need to know at least one card game to get any use from a deck of cards.

The contest was a success, the resulting creations both stretched the capabilities of the system but also the imaginations of designers, who have to deal with reductions in size and abstraction. That various genres and goals can come out of a small playing area and color set is a conceptual marvel not only of the set but of abstracts in general. To compare, stone-placement games based on the Go set are either connection games or area games.

Sadly, after this competition, only a handful of new games popped up on BGG, and the synthesized games are yet to appear; the project itself put on hold. Fortunately, with the launch of Browne's project Ludii, new games may be on its way, and maybe, just maybe, we will see what machine-generated innovations will appear.

In 2020, Emil Danielsen presented nine new games of his own making for the set, after a lull of four years. Not restricting himself merely to abstract fare, Danielsen also approaches the set as a physical object and adds a dexterity game and a game that doesn't even require the board. It might annoy some that such games be included (although one Challenge entry is also a dexterity game, that one focused on ball-rolling and needed a slope), but thinking beyond the system's limits is something humans have excelled in doing; computers can only work within the rules they are given.

Clever yet underlooked, the Shibumi set was made to challenge computers, but it's yet to complete its first task of challenging us.

Quick Essay: Bidding

After the development of the modern trump suit, bidding was the next biggest addition to card games that has led to many great games and expanded possibilities especially for trick-taking games. The great game that has created such possibilities is Ombre, a game not played as often today but its influence resonates every time a contract is bid.

This article will detail with the ways contracts are bid, along with a short traipse through ways trumps are chosen.

Since the creation of the Tarot deck, the idea of a suit that is more powerful than the rest has resonated to the people. 78 cards is a costly affair so players have emulated the same power by randomly assigning a suit that will take the role of the trump suit. This is usually done by turining a card up, otherwise someone does the deed of choosing the suit of privilege based on the information he has.

Games of the latter type tended to create heirarchies of suit choice that affect the points at stake, usually as a multiplier or a base score indicator. Preference of suits are also in the bids themselves, implying a bid of a value renders those below it impossible.

In the context of Tarot games without bidding, dealer tends to get privileges, usually in the form of drawing extra cards into his hand and the subsequent discards counting into his pile. Bidding games eventually came into the scene, and the bidder only needs to win the majority of points, usually alone. Contracts usually differ in how restricted the soloist can play, from reduced talon privileges to giving the opposing team a headstart in points. We're tackling the whole Tarot subject as while it took in a good amount of influences, some elaborations only seem to have stuck in such games, we'll revisit these later.

Games of such sort came to be a bit later on the Tarot timeline, thanks to the game of Ombre, a plain-trick game where players bid for the opportunity to win more tricks than the other players, and the bids mainly differ in whether the soloist can exchange cards from the talon.

The later game of Quadrille then gave rise to changing alliances depending on a called card, the holder's identity a secret. This bidding carried on to games such as whist, where bidding categories turned from trying to win tricks a specific way as in Boston, to bidding for a specific number of tricks, as the game that will eventually become Bridge has.

The above section, but less messy

The goal of a contractor is to fulfil a contract, while defenders try to set it. As simple as that sounds, an asymmetry must take place, with the contractor given some advantage over the defenders in some cases; defenders have their strength in numbers.

Now a quick delineation between bids and feats. Some games require a bid to be made, but along with the contract a number of feats can be announced. A bid contract is the main game that the bidder must play and accomplish. Feats are extra contracts that are played along with the main game but are settled independently and do not affect the nature of the bid.

In some cases feats take place when the condition itself is met, whether or not it is announced, this usually affects the bonus. After the tricks are played, the bid is first assessed then the feats, the latter either affecting the main bid or working independent of it.

Bids in these games can be categorized on how precise the bid can be. Early bids were simply a bid to become the soloist who must now take the majority of points while the remaining players team up against him. The first elaborations are how much default advantages the soloist can surrender in exchange for a chance at a higher value game. In early talon games this meant that the highest bid is reneging the possibility of improving his hand by exchanging cards. 

Preference of suits have appeared in some games usually as a multiplier of a contract. A game of a specific suit as trump sets the value of a contract if won or lost. In games suchas Preference or Skat bidding of this kind means that the available suit games for the bidder shrinks with each higher bid.

For Whist, the change from randomly choosing the trump suit to the ability to select it has given way to interesting complications arising from the naturally simple means of selection. 

From giving the suit declarer contractor responsibilities the next step came in the form of bidding, in this case a responsibility to win the majority of tricks given the previlege of naming the trump suit. Later on the contracts became more precise, from more elaborate contracts to specific minimums to win.

Regarding feats, these extra contracts are announced in conjunction with a bid and affect scores either independently or affecting the bid's score. Achieving or failing feats do not affect the status of the bid, but in some cases involves more scores than the bid itself.

A more novel development is exact bidding, wherein an exact value must be wagered and scores depend on whether the goal is hit or not, with varying degrees of leeway.

Monomatch card inventories

3 items per card

1,2,4

1,3,7

1,5,6

2,3,5

2,6,7

3,4,6

4,5,7

4 items per card:

1,2,3,10

1,4,7,11

1,5,9,12

1,6,8,13

2,4,9,13

2,5,8,11

2,6,7,12

3,4,8,12

3,5,7,13

3,6,9,11

4,5,6,10

7,8,9,10

10,11,12,13

1,2,5,7

1,3,10,11

1,4,6,13

1,8,9,12

2,3,6,8

2,4,11,12

2,9,10,13

3,4,7,9

3,5,12,13

4,5,8,10

5,6,9,11

6,7,10,12

7,8,11,13

5 items per card

1,2,5,15,17

1,3,8,9,12

1,4,14,16,21

1,6,7,10,20

1,11,13,18,19

2,3,6,16,18

2,4,9,10,13

2,7,8,11,21

2,12,14,19,20

3,4,7,17,19

3,5,10,11,14

3,13,15,20,21

4,5,8,18,20

4,6,11,12,15

5,6,9,19,21

5,7,12,13,16

6,8,13,14,17

7,9,14,15,18

8,10,15,16,19

9,11,16,17,20

10,12,17,18,21

6 items per card

1,2,4,11,15,27

8 items per card

1,2,3,4,5,6,7,50

1,8,15,22,29,36,43,51

1,9,17,25,33,41,49,52

1,10,19,28,30,39,48,53

1,11,21,24,34,37,47,54

1,12,16,27,31,42,46,55

1,13,18,23,35,40,45,56

1,14,20,26,32,38,44,57

2,8,21,27,33,39,45,57

2,9,16,23,30,37,44,51

2,10,18,26,34,42,43,52

2,11,20,22,31,40,49,53

2,12,15,25,35,38,48,54

2,13,17,28,32,36,47,55
2,14,19,24,29,41,46,56

3,8,20,25,30,42,47,56

3,9,15,28,34,40,46,57

3,10,17,24,31,38,45,51

3,11,19,27,35,36,44,52

3,12,21,23,32,41,43,53

3,13,16,26,29,39,49,54

3,14,18,22,33,37,48,55

4,8,19,23,34,38,49,55

4,9,21,26,31,36,48,56

4,10,16,22,35,41,47,57

4,11,18,25,32,29,46,51

4,12,20,28,29,37,45,52

4,13,15,24,33,42,44,53

4,14,17,27,30,40,43,54

5,8,18,28,31,41,44,54

5,9,20,24,35,39,43,55

5,10,15,27,32,37,49,56

5,11,17,23,29,42,48,57

5,12,19,26,33,40,47,51

5,13,21,22,30,38,46,52

5,14,16,25,34,36,45,53

6,8,17,26,34,36,45,53

6,9,19,22,32,42,45,54

6,10,21,25,29,40,44,55

6,11,16,28,33,38,43,56

6,12,18,24,30,36,49,57

6,13,20,27,34,41,48,52

6,14,15,23,31,39,47,52

7,8,16,24,32,40,48,52
7,9,18,27,29,38,47,53
7,10,20,23,33,36,46,54
7,11,15,16,30,41,45,55
7,12,17,22,34,49,44,56
7,13,19,25,31,37,43,57

7,14,21,28,35,42,49,52

8,9,10,11,12,13,14,50
15,16,17,18,19,20,21,50
22,23,24,25,26,27,28,50
29,30,31,32,33,34,35,50
36,37,38,39,40,41,42,50
43,44,45,46,47,48,49,50
50,51,52,53,54,55,56,57

1,2,4,14,33,37,44,53

2,3,5,15,34,38,45,54

3,4,6,16,35,39,46,55

4,5,7,17,36,40,47,56

5,6,8,18,37,41,48,57

6,7,9,19,38,42,49,1

1,5,6,18,20,26,27,36

1,8,11,17,19,31,32,36

9 items per card

1,3,6,10,16,24,35,36,52

1,2,13,21,27,31,34,36,58

Shibumi Summaries

This is a list of summarized rules for some games that have been created for or can be played on a Shibumi set for future review purposes. These summaries assume that the basic definitions related to the set are not yet known by the reader and would use more intuitive terms, whether they correlate to the definitions or not

Games included in the Shibumi Rulebook

Spline:

2 players. On an empty board players place a ball of their color on any point. The first player to create a straight line of their color spanning from edge to edge relative to its level wins.

Spline+:

2 players. On an empty board players place either a ball of their color on any point, or they can move a ball of their color to another point that is not above any ball that dropped due to removing said ball. The first to create a straight line of their color spanning from edge to edge relative to its level wins, in case of two opposing lines being formed, the longer line or the mover wins.

Splice:

2 players. On an empty board players either place a ball of their color on a hole or on top of four adjacent balls that has at least one red ball or a red ball on any hole or any four adjacent balls. The first player to create a straight line of their color with at least one red ball spanning from side to side relative to its level wins, if a red ball creates two opposing lines the mover wins.

Spree:

2 players. On an empty board, players hand a ball for their opponent to play on any point. Players cannot hand the same color ball twice. A player who first makes a straight line of either white and red or black and red balls spanning from edge to edge relative to its level wins, pure white or black lines also counting.

Spava:

2 players. On an empty board, players either place a ball of their color or a red ball on any point. Creating a straight line of one's color spanning from edge to edge relative to its level wins, but creating a line at least two balls long shorter than a winning line loses.

Splade:

3 players. On an empty board, players place a ball of their color on any point. The game ends if a straight line made of two colors spanning from edge to edge relative to its level is made. The owner of the color on the line that moves later wins.

Sparro:

2 players. On a board with four neutral balls on the center four holes and another on top of them, players place a ball of their color on any point. If the first ball is placed on a corner, the next ball must not be placed on another corner and vice-versa. The game ends after  players have each placed 12 balls of their color. Whoever has the most straight lines of three balls either lying flat or climbing upward wins, if tied, the last player to create a line of three wins, otherwise the latter player.

Sploof:

2 players. On a board with 12 red balls on the edge holes, players start with a stock of two balls of their color. Players either place a ball from the stock on a point or remove a red ball and adding two more balls to their stock. The first player to create a straight line of four connected balls when viewed from above wins.

Spaniel:

2 players. On an empty board, players place two balls on any two points in this order:

Player 1: Black-White
Player 2: Red-Black
Player 3: White-Red

The first player to create a visible straight line of three adjacent balls wins.

Span:

2 players. On an empty board, players place a ball of their color on any point. White wins if he makes a visible connection from above from left to right, while Black wins if he makes a visible connection from above from top to bottom.

Sponnect:

2 players. On a board with four red balls on the center four squares and another red ball above them, players place a ball of their color on any point. White wins if he makes a visible connection from above from left to right, while Black wins if he makes a visible connection from above from top to bottom.

Spight:

2 players. On a board with white and black balls placed in a checkerboard setup, players move a ball of their color with at most one other ball above it in a knight move on the same level or upwards a level, i.e. onto the nearest four-ball platform.  The first player to connect all his balls into one group wins.

Spice:

2 players. On a board with four red balls on the center four squares and another red ball above them, players place one of two of their balls. If a move creates a larger group of balls than before, the mover's opponent can remove any movable ball on their next turn. The game ends when the pyramid is complete, after removing the top ball the owner of the largest group seen from above wins.

Pylos:

2 players. On an empty board, players each have a stock of 15 balls of their color and may place them on any hole. When four balls form into a square, a player can either place a ball from their stock onto it or move one of their balls not underneath any ball onto the point. If a player makes a square out of their own color, they can take back any two balls of their color back to their stock. The player who completes the pyramid wins.

The Pyramid of the Flying Marbles: 

3 players. On a board randomly filled with 10 white, black and red balls, players capture marbles based on turn order, starting with whoever owns the topmost ball. White captures red which then captures black. Captures can only be between balls that touch and can be done on balls that support at most one ball, this is done by removing the target and moving the capturing marble on its place. Balls can drop down after capture. If a player can't capture, they can pass. Game ends after all players pass, the winner being the one who has the most balls remaining.

Chess Piece Essays: A quick rundown

This is an introduction to my attempts at a sort of reclassification of chess pieces. To be honest, I am merely working on some older works of chess piece taxonomy, so I am not going to start from the ground up unless it's a blank that needs filling. 

Usually, all this piece talk is within the context of either fairy/generalized chess or chess variations, but let's assume you have no idea what chess is at all. What makes a piece?

The Humble Piece

A chess piece is a piece that sits on a chessboard square that has a specified vector of movement across the board. This vector dictates which other squares it can go to from the square it is currently on. 

For a chess piece to move, it has to leave the square it sits on and move to one of the possible destination squares. This can be blocked by other pieces or the geography of the board. If a piece is unable to move it does not cease being a piece.

While any Tom, Dick and Harry can place rules of movement for a piece, these essays deal with the ones we can neatly classify. This makes the subject matter a bit of a list of the ways a piece can move in a more or less regular manner.

The Way They Move

If it can only move one square at a time, it's a leaper, if it can make multiple leaps, it's a rider, if its movement in any way requires another piece, it's a hopper. Pieces that combine two or three of these move types are hybrids and are strictly only seen in chess variations.

Assume a distance, if a piece can go in every direction allowed by the distance, then it has full range, otherwise it is limited. E.g. if a piece can move a certain way in one direction but a different way in another, then this piece's range is limited.

Generally, a piece is blocked from a destination square if an allied piece is on it. Depending on the piece it can even block further destination squares.

If an opponent piece is on a destination square, the piece can go to that square and displace the opposing piece, thereby capturing it and causing a stop to the piece's trajectory. There are other ways of capture, but that's a different can of worms so captures in these essays are implied to be displacement captures. 

Some chess variations depend on a piece whose capture can signify a win or loss of specific pieces. In orthodox chess it's the king that must be captured to win. Even if the action of capturing is itself not done, some moves cannot be done upon the implication that it will cause the capture of the king on the next turn. Other pieces can exhibit these same properties and in some cases extra provisions are needed for some class of pieces.

What to Expect

This is simply a sort of amateur attempt to give a consistent taxonomy of pieces that are used in generalized chess. Any names used for some pieces are either based on common parlance or the closest available naming that I can find, creating new names simply a last-ditch effort. As this was initially made as a text-only piece, pictures will be added later but the text won't be edited to reference any diagrams.