Bent Riders

By Ralph Betza

One of the myths about the game of Go is "the rules are so simple compared to chess that if we ever meet an extraterrestial intelligence they will surely have discovered the game themselves".

Studying the values of chess pieces has shown me that the game of Chess embodies such basic geometrical concepts that if we ever meet the ETs, we will find they have a game with pieces that move, and there will be a Rook, a Bishop, a Nightrider, and perhaps even, if their civilization is sufficiently advanced, a Crooked Bishop.

In H.J.R. Murray's History of Chess, page 181 states that the Alfonso manuscript was published in about the year 1283 which describes the move a piece called the Gryphon, a piece I loved at first sight. The move description could be translated as "A move compounded of one step diagonally, followed by any number straight." [Note 1]

In the same place was described a piece called the Unicornio (medieval Spanish, but actually describing a Rhinoceros), moving as Knight then Bishop. Supposedly the Unicorn couldn't make a capture using its Knight move, but I'll ignore that silly rule. [Note 2]

Not described there is a piece which makes a one step Rook move and then continues outwards as a Bishop. For lack of a name, I'll call it the Aanca (13th century Spanish for "Gryphon"). Although the Aanca is not described, one can suppose that the same mind who conceived the Gryphon and the Unicorn probably also considered the Aanca. [Note 3]

All three pieces are expressions of the same geometrical idea -- they are "bent riders", that move one step in one direction and then continue in a different direction.

On page 181, Murray also describes a variant from India, written in Persian and dated circa 1796, said to have used algebraic notation, and to have included the modern Bishop and Queen, as well as the Chancellor (Rook plus Knight), the Archbishop (Bishop plus Knight), and even the Amazon (Queen plus Knight). These are basic geometrical constructs, and as you see they are not new. The ETs would have these ideas as well. Note 4]

On the 8x8 board, riders have been formed from the moves of W (Wazir), F (Ferz), D (Dabbabah), A (Alfil), and N (Knight), and therefore there are 25 possible basic forms of "bent riders" that might be useful on an 8x8 board -- only a few of which are known from history.

Value of a Bent Rider

Consider first the case of the Left Gryphon, which makes an F move, turns 45 degrees left, and continues as a Rook. From e1 it could go to f2, f3, ..., f8, or to d2, c2, b2, a2. If its first step is on the board, it has the same mobility as a Rook, simply shifted one square sideways. Because the probability of a destination square being on the board is ((w-x)*(h-y)/(w*h), the average mobility of the Left Gryphon is precisely 7/8 of a Rook.

By the same reasoning, the Left Aanca "is worth" 8/7 of a Bishop, plus a bonus for not being colorbound.

Riders are different than jumpers, and so treating the F and W as having different values when calculating a rider's average mobility does not violate the equivalence of atoms proposed by my theory of ideal values. At least I think so.

This valuation is simple and elegant, based simply on geometry: the ratio of the mobility of the first step determines the relative value of a bent rider and its corresponding straight rider. Compared to the laborious calculation needed for the Crooked Bishop, this is extremely pleasant. The fact that the result is so geometrically precise is also pleasant in a field where all the other numbers are hedged about with doubt.

However, the full Gryphon is not worth 14/8 of a Rook because its first step is shared. Things are still simple, though: the first step of a Rook is worth a third of a Rook, because the R is worth 3 atoms. Thus, the full Gryphon has the mobility of 5/3 times 7/8 Rook, 1.46 Rooks. (Earlier, I had said 1.42 Rooks, which is such a very small difference that there is no sense worrying about it.)

The Bishop's first step is worth half, so the full Aanca is worth 3/2 times 8/7 Bishops, 1.7 Bishops. That's 3.4 atoms, while the Gryphon is worth 4.4 atoms; and because the Aanca is not colorbound but the Bishop is, the expected practical value of the Aanca might be about 1.9 Bishops.

A piece combining the moves of Gryphon and Aanca would intersect at the (1,2) squares and so one would also need to subtract (2 * magicnumber * onboard) and add ((1-((1-magicnumber)**2) * onboard). In the simplified calculation, we'd use 2 standard atoms as the value of onboard; (.91*2)-(1.4*2), subtract one full atom, and the combined Gryphaanca is 6.8 atoms compared to 5 for the Q, or 7 for the Amazon.

I presume that the Unicorn cannot jump from e1 to f3 and then continue as a Bishop to e4, d5, c6, b7, a8; instead, from e1-f3-g4-h5. If this is correct, its value is two Bishops times the ratio of the mobility of a single F move and a single N move, exactly 12/7 of a Bishop: 1.7 Bishops, but when you correct for colorboundness the Unicorn is worth very nearly as much as 2 Bishops.

Now you know how to calculate an expected value for any bent rider that does not cross its center line.

Crossing Bent Riders

A piece that moves one square as Ferz, then turns 90 degrees and continues as Bishop is more difficult to value. Its first four steps are exactly the path a Zigzag Bishop would follow.

I am not in the mood to figure out its value right now. The method has been explained should the reader wish to do so.

Twenty Five Bent Riders

How lucky you are that I shall not name them all! Here are a few that I find especially interesting.

A piece that makes an F move followed by an outwards Dababbah-Rider move looks interesting, and the Aviaanca would be a piece that goes W followed by AA.

W then NN, presumably if it goes from e1 to e2 it can then continue to d4 or f4 but not c3 or g3 -- although both flavors are interesting!

N then WW, again from e1 to f3 continues via f4 but not g3; and in this case the alternate form seems much too powerful.

If you count the Rhino pieces, there are more possible riders, and a piece that makes a W move followed by a Crooked Bishop move is also not among the "25 possible" bent riders.

Original Bent Riders

Here are a few good bent rider pieces that do not fit exactly into the scheme of the "25 possibles".

The Twin Tower

Because the Staunton Rook is made in the shape of a tower, the Twin Tower[1] might be the name for a bent rider which moves one square diagonally and then continues outwards as a Rook, but only forwards or backwards. In other words, if it goes from e1 to d2, its next step can only be to d3; a Gryphon could have gone from e1 to d2 to c2, so we see that the Twin Tower is more or less a half-Gryphon.

Although the Twin Tower "is worth" a mere 7/8 of a Rook, I expect that its practical value will be equal to a Rook -- perhaps even more -- because of its powerful forward influence.

The SnakeTongue and the Vivi

The SnakeTongue is a bent rider which moves forwards or backwards as Wazir then continues outwards as a Bishop -- it is half an Aanca, and so its estimated practical value is about as much as a Bishop, or perhaps a little bit less.

The Vivi moves forwards or backwards as Wazir then continues forwards as a Bishop. That's right, after moving from e2 to e1 it would continue via d2-c3-b4-a5 or via f2-g3-h3! The estimated value is roughly a Bishop and perhaps a bit more because of its forwardness -- but if it needs to retreat it is in big trouble! This is a very charming piece, whose movement pattern looks like a V in a V.

The Annoyance

The Minor Annoyance moves sideways as a Wazir then continue forwards outwards as a Nightrider. It "is worth" 29/32 of a Knight, but I suspect that its practical value is less because of its dispersed power pattern; and besides, it's colorbound in a strange way because if it starts on the first rank it can never get to the second rank!

For example, it could move from e4 to d4 then continue either via c6 to b8 or to c2.

Its difficult development will annoy its owner and its long-range penetrating attack will annoy its opponent.

The Major Annoyance moves as the Minor Annoyance, but also may move or capture one square directly rearwards (that is, as a retreating Wazir). Its value seems to be 33/32 of a Knight, but it is doubtless worth less than Knight.

And What About

What about Ferz then Cannon? Its move has the basic shape of the Gryphon's move, but the difference is enormous. (I'm assuming that when it uses the Cannon part of its move it must leap both to move and to capture; but the Ferz part of its move is pure F, and so it's not simply a "bent Cannon".)

The F move means that it always has some value even when there are no pieces to jump over, but the need to jump makes it much weaker than a Gryphon -- for example it can't go from e1 to f3 in one move.

Summing Up

I had written this a few months ago, but then came one of those times when I didn't feel like logging on to the internet for a few months, and when I simply didn't think about chess variants at all for a while; and what you read now is three times as long as the original.

The Wazir and Rook are simple expressions of basic geometry, but the Rhino and Gryphon are expressions of two basic geometries at once.

I hope that the ability to estimate the values of these second-order pieces will make it easier for people to use them in more chess variants.

Footnotes

These footnotes are written by Greg Strong and describe edits to Betza's original text and provide other information. The original piece contained inaccurate readings of Murray and additional historical insight has been discovered since then by Sonja Musser and Jean-Louis Cazaux, among others.