User:LithiumFlash/sandbox

Testing links to section name in article

Maximum number of givens

Terminology

Other terminology

Fairy-Max

Fairy-Max is a free and open source chess engine which can play orthodox chess as well as chess variants. Among one of its more notable features, is the ability of users to define their own custom variant chess pieces for use in games.

The Fairy-Max module is a chess engine only, but is packaged with the WinBoard, which serves as the graphical user interface. Users can play against the Fairy-Max engine, or play the engine against other engines, or play Fairy-Max against itself, for the purpose of analyzing chess moves, chess variants, or variant chess pieces.

See also

• Chess engine
• ChessV (also plays chess variants)
• List of chess software

Farlow Gap Trail

Farlow Gap Trail
Trailhead (Farlow Gap section)
Length	3.1 mi (5.0 km)
Location	Pisgah National Forest, North Carolina, United States
Trailheads	Cove Creek Davidson River Trailhead
Use	Mountain biking (primary), Hiking (secondary)
Highest point	4,527 ft (1,380 m)
Lowest point	3,313 ft (1,010 m)
Grade	11% (average)
Difficulty	Hiking — Unrated Mountain biking — Double-black (most difficult)
Season	All
Sights	Toms Spring Falls (near start of extended loop)
Hazards	large drops, tree roots, unimproved creek crossings.
Surface	rock, hardpack, gravel.

Farlow Gap is a popular trail for mountain biking and hiking. It is an expert-level trail, and considered "one of the toughest mountain bike trails in Pisgah National Forest." It is primarily a downhill trail, about 3.1 miles (5.0 km) in length. It can be combined with other trails to form a loop of about 14 miles (23 km), and can require anywhere from three to six (or more) hours to complete.

Nearly the entire trail is singletrack, and starts with a relatively flat section but quickly becomes steep and becomes fall line into a rock garden. It has been described as "brutal" and is one of the steepest and rockiest downhill trails in the Pisgah National Forest. Many people wind up riding to the left or right of the rock garden, or dismounting and walking similarly to one side of it. The trail drops about 2000 ft in 3 miles then connects with the Daniel Ridge trail. After the early steep sections, the remainder of the trail includes some shorter uphill sections, other rocky areas, occasional drops, other technical sections, and several creek crossings.

A creek running across Farlow Gap trail. The trail is unimproved, so mountain bikers and hikers find their own path to cross the creek.

A waterfall which can be reached from Daniel Ridge Trail..

Farlow Gap is usually ridden as part of a loop. This requires a ride to the top, which is about 9 miles (14 km) on a gravel doubletrack trail. The last several miles of the climb is the steepest. The end of Farlow Gap links with Daniels Ridge Trail, which can be taken in one of two directions to get back to the starting point. Near the trailhead of Daniel Ridge trail is the Daniel Ridge waterfall.

Other activities

This trail is used mainly by mountain bikers, but can be equally enjoyed by hikers. Due to the steepness of many sections, horseback riding, and sled-dogging is prohibited. Challenges for hikers include rocky sections. Trail rules require mountain bikers to yield to hikers; nevertheless, hikers should be watchful of mountain bike traffic. Mountain bike races sometimes include Farlow Gap along their routes, and hikers should either avoid the trail on these days, or be especially watchful. The trail crosses several creeks where water bottles can be filled.

Animals, plants, and geology

There is a variety of birds, reptiles and mammals in the area. Deer, rabbits and bobcats may be spotted by watchful hikers and mountain bikers.

Photo gallery

• 
  Sign at Farlow Gap trailhead (starting point of downhill bike trail).
• 
  A creek running across the trail.
• 
  A creek running across the trail.
• 
  A small waterfall visible from the trail.
• 
  The Daniel Ridge waterfall (located near the trailhead of Daniel Ridge trail)
• 
  Passers-by create rock balancing sculptures near a trailhead of the Daniel Ridge Trail.
• 
  (in info box).

References

5 categories:

• Pisgah National Forest
• Mountain biking
• Hiking trails in North Carolina
• Trails
• National Recreation Trails of the United States

Cycling

BMX bike

Street trials

26" wheel

gap jump

log ride

log pile

rock garden

skinny

technical

wall-ride

superman

TTF

Rules of the Trail

Cycling infrastructure

link to image:

Windsurfing

• 
  Board
• 
  Boom
• 
  Boom (connected to mast)
• 
  Boom (with harness lines)
• 
  Boom (detail showing connection to mast)
• 
  Harness lines
• 
  A universal joint

Combinatorial game theory

Examples of links:

elastic coupling

Standardized Format for Sudoku Images

"Hawk or other text"

aaa

One of the most studied games is chess, including work by Alan Turing, who wrote, "If one can explain quite unambiguously in English, with the aid of mathematical symbols if required, how a calculation is to be done, then it is always possible to programme any digital computer to do that calculation, provided the storage capacity is adequate," and Claude Shannon, who estimated there are 10⁴³ legal positions in chess. Chess remains unsolved, although extensive study, including work involving the use of supercomputers has created chess end-game tablebases, which shows the result of perfect play for all end-games with seven pieces or less. Infinite chess has an even greater game complexity than chess, unless only limited end-games are studied, or if composed positions with a small number of pieces is being studied.

Combinatorial Explosion

(Sudoku grids of (1×1), 2²(4×4), 3²(9×9), etc.)

n	The number of Latin squares of order n	The number of Sudoku grids of order n (Sudoku boxes √n×√n)
1	1	1
2	2	--
3	12	--
4	576	288
5	161,280	--
6	812,851,200	--
7	61,479,419,904,000	--
8	108,776,032,459,082,956,800	--
9	5,524,751,496,156,892,842,531,225,600	6,670,903,752,021,072,936,960
10	9,982,437,658,213,039,871,725,064,756,920,320,000	--
11	776,966,836,171,770,144,107,444,346,734,230,682,311,065,600,000	--

aaaAAAaaa

Dimensions	Nr Grids	Attribution	Verified?	Rel Err
1x?	see Latin squares			n/a
2×2	288	various	Yes	-11.1%
2×3	28200960 = c. 2.8×107	Pettersen	Yes	-5.88%
2×4	29136487207403520 = c. 2.9×1016	Russell	Yes	-1.91%
2×5	1903816047972624930994913280000 = c. 1.9×1030	Pettersen	Yes	-0.375%
2×6	38296278920738107863746324732012492486187417600000 = c. 3.8×1049	Pettersen	No	-0.238%
3×3	6670903752021072936960 = c. 6.7×1021	Felgenhauer/Jarvis	Yes	-0.207%
3×4	81171437193104932746936103027318645818654720000 = c. 8.1×1046	Pettersen / Silver	No	-0.132%
3×5	unknown, estimated c. 3.5086×1084	Silver	n/a
4×4	unknown, estimated c. 5.9584×1098	Silver	n/a
4×5	unknown, estimated c. 3.1764×10175	Silver	n/a
5×5	unknown, estimated c. 4.3648×10308	Silver / Pettersen	n/a

Sudoku Text

A paper by Gary McGuire, Bastian Tugemann, and Gilles Civario, released on 1 January 2012, explains how it was proved through an exhaustive computer search that the minimum number of clues in any proper Sudoku is 17, and this was independently confirmed in September 2013.

Main article: Sudoku

 Main articles: Sudoku, Sudoku solving algorithms

Sudoku Templates

References:

http://www-imai.is.s.u-tokyo.ac.jp/~yato/data2/MasterThesis.pdf

http://theinf1.informatik.uni-jena.de/publications/sudoku-weller08.pdf

https://www.cs.wmich.edu/elise/courses/cs431/icga2008.pdf

http://geevi.github.io/2014/puzzles.html

An example is here: User_talk:David_Eppstein

Matching Cell Triplets

1 2 3 4 5 6 7 8 9

7 8 9 1 2 3 4 5 6

4 5 6 7 8 9 1 2 3

Header 1	Header 2	Header 3
row 1, cell 1	row 1, cell 2	row 1, cell 3
row 2, cell 1	row 2, cell 2	row 2, cell 3

1	2	3
4	5	6
7	8	9

1	2	3
4	5	6
7	8	9

A paper released on 1 January 2012 proved through exhaustive computer search that the minimum number of givens is 17 in general Sudoku. A few 17-clue puzzles with diagonal symmetry were provided by Ed Russell, after a search through equivalence transformations of Gordon Royle's database of 17-clue puzzles. [1] Sudoko puzzles with 18 clues have been found with 180° rotational symmetry, and others with orthogonal symmetry, and it is also not known if this number of clues is minimal in either case.[2] Sudoko puzzles with 19 clues have been found with two-way orthogonal symmetry, and again it is unknown if this number of clues is minimal for this case.[4] At least one sudoku with symmetry on both orthogonal axis, 90° rotational symmetry, 180° rotational symmetry, diagonal symmetry, and is also automorphic, has been found. Again here, it is not known if this number of clues is minimal for this class of sudoko.

Sudoku

A range of clue positions insufficient for a proper Sudoku

Sudoku with 30 cell (5 x 6) empty rectangle

Sudoku with nine empty groups

aaa

A Sudoku with 17 clues.

A sudoku with 17 clues and diagonal symmetry.

"Raphael" - A sudoku with 18 clues and orthogonal symmetry.

An automorphic Sudoku with 24-clues, and complete geometric symmetry.

"Tourmaline" - A Sudoku with 19-clues and two-way orthogonal symmetry.

A paper released on 1 January 2012 proved through exhaustive computer search that the minimum number of givens is 17 in general Sudoku. A few 17-clue puzzles with diagonal symmetry were provided by Ed Russell, after a search through equivalence transformations of Gordon Royle's database of 17-clue puzzles. Sudoko puzzles with 18 clues have been found with 180° rotational symmetry, and others with orthogonal symmetry, and it is also not known if this number of clues is minimal in either case. Sudoko puzzles with 19 clues have been found with two-way orthogonal symmetry, and again it is unknown if this number of clues is minimal for this case. At least one sudoku with symmetry on both orthogonal axis, 90° rotational symmetry, 180° rotational symmetry, diagonal symmetry, and is also automorphic, has been found. Again here, it is not known if this number of clues is minimal for this class of sudoko.

Standardized Format for Sudoku Images

S1 - Suggested standard

S2 - Lead image (1 of 2) - "A Sudoku Puzzle"

S3 - Lead image (2 of 2) - "And its Solution"

"A"
(for reference only)

S4

S5

S6

S7

S8

Article(s)

Sudoku, Mathematics of Sudoku, Sudoku solving algorithms, Glossary of Sudoku

Links to Useful Templates

Template for SVG Icons: [SVG chess pieces]

Chess diagram template: [Chess diagram]

Partial results

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

A mate-in-546 position found in the Lomonosov 7-piece tablebase. White to move. (In this example an 8th piece is added with a trivial first move capture).

Endgame tablebases have solved chess to a limited degree, determining perfect play in a number of endgames, including all non-trivial endgames with no more than seven pieces or pawns (including the two kings).

One consequence of the 7-piece endgame tablebase, is that many interesting theoretical chess endings have been found. One such example is a "mate-in-546" position, which with perfect play is a forced checkmate in 546 moves.

Such a position is beyond the ability of any human to solve, and indeed no chess engine plays it correctly (unless the engine itself is designed to have access to the endgame tablebase).

Finding positions such as this can make one speculate what other interesting chess situations will be found, as more chess positions are solved. Unfortunately, adding only one single new piece to a chess position expands the complexity of the game-tree by such a vast amount, that the development of an 8-piece endgame tablebase is currently considered an intractable problem.

Relative Value of Pieces

While a large amount of information can be found concerning the relative value of variant chess pieces, there are few resources where it is in a concise format for more than just a few piece types. One challenge of producing such a summary is that piece values are dependent upon the size of boards they are played on, and the combination of other pieces on the board.

On an 8×8 board, the standard chess pieces (pawn, knight, bishop, rook, and queen) are usually given values of 1, 3, 3, 5, and 9 respectivelly. When the basic pieces wazir (W), ferz (F), and mann (WF), are played with a similar mix of pieces, they are typically valued at around 1.2, 1.5, and 3.2 points respectivelly. Three popular compound pieces, the archbishop (BN), chancellor (RN), and amazon (QN) have been estimated to have point values around 8, 9, and 11.5 respectivelly. (Due to the powerful ability of the three later pieces, it is uncommon for more than one to be played on a standard 8x8 board).

Apart from these, reliable estimates are not be well established for many other pieces. Even when the same game format is assumed (board size and combination of other pieces), there is often little agreement on the specific value of many other pieces. Compound pieces are sometimes approximated as the sum of their component pieces, or estimated to be slightly higher due to synergestic effects (such as it is for the archbishop and chancellor).

For purely jumping pieces (including moves of a single square), one formula has been developed, sometimes good-naturedly called "Muller's Short Range Leaper Law" which estimates a pieces value as:

$value=\alpha \left(30.0(N)+0.625(N)^{2}\right)$, centipawns

where:

$\alpha$ = factor based on scale for other pieces (1.0: classical, i.e. knight = 300; 1.1: Kaufman, i.e. knight = 325)

$N$ = squares attacked

Although regarded as reliable, this formula is limited in that it only applies to leapers jumping not more than two squares. Furthermore, it doesn't take into account some factors which may influence a piece's value, such as the distance of jumps (shorter jumps are usually worth more), and direction (pieces favoring forward captures are typically worth more than symetrically capturing pieces).

For sliding pieces, and sliding-leaping compounds, there seems to be no corresponding formula to estimate piece values. It is generally presumed that with bigger boards, sliding pieces (i.e bishop and rook) generally become more valuable relative to short-range jumpers, if for no other reason that they can travel about the board more quickly.

In Context of game theory

There is a consensus among the math community that infinite chess is decidable; that is, given a position (such as $\mathbb {Z} \times \mathbb {Z}$, and assuming pawns do not promote) and with a finite number of chess pieces which are uniformly mobile and with constant and linear freedom, and (for example) white to move, there is an algorithm that will answer if white can win or force a draw, against any defense by black.

From:

Is infinite chess played using standard chess pieces decidable? In other words, in a game of infinite chess, given a position (such as $\mathbb {Z} \times \mathbb {Z}$, and assuming pawns do not promote) with a finite number of standard chess pieces, and (for example) white to move, is there an algorithm to determine if white can win, or force a draw, against any play by black.

More generally this strategy works for any chess-like game in which the pieces can be classified into "ultra-mobile" and "para-mobile" types with constant and linear freedom respectively. For example it is also true for checkers because all pieces are para-mobile.

$\mathbb {Z} _{3}\oplus \mathbb {Z} _{3}$

$\mathbb {Z}$

$\times$

$\mathbb {Z} \times \mathbb {Z}$

${\mbox{Number of Grids}}\simeq {\frac {b_{R,C}^{C}\times b_{C,R}^{R}}{(RC)!^{RC}}}$

Testing Math Equations

(Sudoku grids of (1×1), 2²(4×4), and 3²(9×9))

(n = 9 is the commonly played 9×9 Sudoku)
(Sudoku does not include grids where √n is irrational)
A Sudoku is usually n×n with boxes of size√n×√n. (and n² cells).
(n is not included in table when √n is irrational)

√n×√n

(and n² cells)

√n×√n

(and n² cells)

n×n

aaa

f(x) = x²

value = 1.1*N*(30+5/8*N). (plain)

value = 1.1*N*(30+5/8*N). (italics)

value = 1.1*N*(30+5/8*N). (bold)

value = α((30.0(N) + 0.625(N)^2), centipawns. (plain)

value = α(30.0(N) + 0.625(N)²), centipawns (math - HTML)

value = α(30.0(N) + 0.625(N)²), centipawns (math - HTML, italics)

$value=\alpha \left(30.0(N)+0.625(N)^{2}\right),centipawns$ (math - LaTeX)

$value=\alpha \left(30.0(N)+0.625(N)^{2}\right)$, centipawns (math - LaTeX)

more math - LaTeX:

$value=\pi \left(-r^{3}+{\frac {r^{3}}{3}}\right).$

$V=\pi \left[r^{2}x-{\frac {x^{3}}{3}}\right]_{-r}^{r}=\pi \left(-r^{3}+{\frac {r^{3}}{3}}\right)={\frac {4}{3}}\pi r^{3}.$

$V={\frac {4}{3}}\pi r^{3}.$. (volume of a sphere)

$\scriptstyle {\frac {64!}{32!{8!}^{2}{2!}^{6}}}$

${\frac {64!}{32!{8!}^{2}{2!}^{6}}}$

(${64!}$)/(${32!{8!}^{2}{2!}^{6}}$)

10⁴³

Shannon also estimated the number of possible positions, "of the general order of $\scriptstyle {\frac {64!}{32!{8!}^{2}{2!}^{6}}}$, or roughly 10⁴³". This includes some illegal positions (e.g., pawns on the first rank, both kings in check) and excludes legal positions following captures and promotions.

Shannon also estimated the number of possible positions, "of the general order of ${\frac {64!}{32!{8!}^{2}{2!}^{6}}}$, or roughly $10^{43}$, 10⁴³". This includes some illegal positions (e.g., pawns on the first rank, both kings in check) and excludes legal positions following captures and promotions.

The Shannon number, named after Claude Shannon, is a conservative lower bound (not an estimate) of the game-tree complexity of chess of $10^{120}$, based on an average of about $10^{3}$possibilities for a pair of moves consisting of a move for White followed by one for Black, and a typical game lasting about 40 such pairs of moves.

Shannon's Calculation

Shannon showed a calculation for the lower bound of the game-tree complexity of chess, resulting in about $10^{120}$possible games, to demonstrate the impracticality of solving chess by brute force, in his 1950 paper "Programming a Computer for Playing Chess". (This influential paper introduced the field of computer chess.)

Shannon also estimated the number of possible positions, "of the general order of ${\frac {64!}{32!{8!}^{2}{2!}^{6}}}$, or roughly $10^{43}$". This includes some illegal positions (e.g., pawns on the first rank, both kings in check) and excludes legal positions following captures and promotions. Taking these into account, Victor Allis calculated an upper bound of $5\times 10^{52}$ for the number of positions, and estimated the true number to be about $10^{50}$. Recent results improve that estimate, by proving an upper bound of only 2¹⁵⁵$2^{155}$, which is less than $10^{46.7}$and showing an upper bound $2\times 10^{40}$ in the absence of promotions.

Allis also estimated the game-tree complexity to be at least $10^{123}$, "based on an average branching factor of 35 and an average game length of 80". As a comparison, the number of atoms in the observable universe, to which it is often compared, is estimated to be between $4\times 10^{79}$ and $4\times 10^{81}$.

Number of Sensible Chess Games As a comparison to the Shannon number, if chess is analyzed for the number of "sensible" games that can be played (not counting ridiculous or obvious game-losing moves such as moving a queen to be immediately captured by a pawn), then the result is closer to around $10^{40}$ games. This is based on having a choice of about three sensible moves at each ply (half a move), and a game length of 80 ply (40 moves).

Number of Sensible Chess Games

If chess is analyzed for the number of "sensible" games that can be played (not counting ridiculous or obvious game-losing moves such as moving a queen to be immediately captured by a pawn), then the result is closer to around 10⁴⁰ games. This is based on having a choice of about three sensible moves at each ply (half a move), and a game length of 80 ply (40 moves).

guard

hawk

huygens

Infinite chess

Trappist-1

Introduction to Fairy Pieces

Fragment of a chessboard and chess pieces, Russia, XVII century (1601-1672). This may have once have been the "standard" form of chess.

Today's chess exists because of variations someone made to the rules of an earlier version of the game. The queen we use today was once able to move only a single square in a diagonal direction, a ferz. Today, this piece still starts next to the king, but has gained new movement and is today's queen. Thus, the ferz is now considered a non-standard chess piece. Chess enthusiasts still often like to try variations of the rules and the way pieces move. Piece which move differently from today's standard rules are called "variant" or "fairy" chess pieces.

Compound pieces combine the powers of two or more pieces. The archbishop, chancellor, and amazon are three popular compound pieces, combining the powers of the minor pieces from orthodox chess. The icon of a compound piece is often itself a compound of its components.

Archbishop (bishop + knight)

Chancellor (rook + knight)

Amazon (queen + knight)

Marine pieces are a compound pieces consisting of a rider (for ordinary moves) and a locust (for captures) in the same directions. Marine pieces have names alluding to the sea and its myths...

Disks representing the advisor of xiangqi (red and black pieces, front and back).

Keima
(the knight)

Hisha
(the rook)

Game pieces of shogi.

Some classes of pieces come from a certain game, and will have common characteristics. Examples are the Chinese pieces from xiangqi, a Chinese game similar to chess. The most common Chinese pieces are the leo, pao and vao (derived from the Chinese cannon) and the mao (derived from the horse). Those derived from the cannon are distinguished by moving as a hopper when capturing, but otherwise moving as a rider. Less frequently encountered Chinese pieces include the moa, nao and rao.

Pieces from xiangqi are usually circular disks, labeled or engraved with a Chinese character identifying the piece type. Pieces from shogi and taikyoku shōgi (Japanese games similar to chess) are usually wedge-shaped chips, with Japanese characters identifying the piece type.

Special attributes

Fairy pieces vary in the way they move, but some may also have other special characteristics or powers. The joker (in one of its definitions) mimics the last move made by the opponent. So for example, if white moves a bishop, black can follow by moving the joker as a bishop. Another piece, which doesn't move in a special way, but has a special characteristic is the bulldog. This piece moves just as a pawn, and usually starts on the same rank as the pawns, but it is transparent to pieces of its own color. The transparency means pieces can move right past the bulldog (so for example, a rook can move past its bulldog and immediately move into play on the first move of a game).

Images for Chess and Chess Diagrams

A checker (pink background):

A checker (transparent background):

A checker king (transparent background):

A typical icon of the grasshopper:

An illustration of the alfil:

A typical icon of the lion:

A typical icon of the camel:

The elephant as depicted in Persian chess:

A typical icon of the knightrider:

A typical icon of the amazon:

A typical icon of the amazon:

A typical icon of the archbishop:

A typical icon of the bulldog:

A typical icon of the siege tower:

A typical icon of the champion:

A typical icon of the chancellor:

Typical icons of the guard:

A typical icon of the wizard:

A typical icon of the fool:

Testing Images within Text (Blue Backdrop)

A typical icon of the amazon:

A typical icon of the boat:

A typical icon of the elephant:

A typical icon of the wizard:

A typical icon of the unicorn:

A typical icon of the giraffe:

A typical icon of the zebra:

A typical icon of the fool:

A typical icon of the joker:

Chess Piece for Diagrams

The names of the pieces are those given in algebraic notation:

• k = king
• q = queen
• r = rook
• b = bishop
• n = knight
• p = pawn

There are also some fairy chess pieces available:

• a = archbishop (knight+bishop compound)
• c = chancellor (knight+rook compound)
• f = inverted king
• g = inverted queen
• m = inverted rook
• j(e) = inverted bishop
• N(s) = inverted knight
• h = inverted pawn
• z = champion (for Omega Chess)
• w = wizard (for Omega Chess)
• t = fool (for Omega Chess)

Description

In its most basic form, the huygens jumps any prime number of squares (2, 3, 5, 7, 11, 13...) in orthogonal directions, but in game rules it is usually restricted to jumps of five or more (5, 3, 7, 11, 13...). The minimum restriction is added to make the huygens a long jumper rather than a close-range attacker. It is used on large or unbounded gameboards. The fact that it makes jumps of prime numbers introduces some interesting features to the piece, and the games it is used in.

Since it jumps prime numbers of squares, other pieces that jump such as the knight, dababba, and hawk, are required to make an innefficient manuever when chasing the huygens. Since a prime number is not a multiple of any other number (other than 1 and itself), another jumper which attacks a huygens cannot do so again unless it makes a non-full-length jump, or moves in a path which is not the straightest path towards the huygens.

The huygens also leads to another interesting aspect in the study of chess. Mathematical studies have shown that the game of chess is decidable, both when played on a normal 8×8 board, and also an unbounded gameboard. Although neither of these games has been solved, there does exist a decision tree with finite depth and finite vertices that will show whether the game is a draw, a win for white, or a win for black. However, it has been suggested that by adding the huygens to the set of chess pieces, this conclusion needs to be re-examined. Since the huygens jumps prime numbers of squares, and the complete set of prime numbers is unknown, the game of chess when played with the huygens on an infinite gameboard may be undecidable. That is, any algorithm or mathematical method to solve the game cannot be completed, because it requires using the set of prime numbers, and this set is unknown, and there is no method to compute it. Nevertheless, despite this distinction, the huygens by itself does not change the game very much in a practical way (other than how the game is changed with the introduction any other variant chess piece). Whether the Huygens is used or not, the game of chess, especially when played on an infinite chessboard, is complex enough that the difference is not noticed in normal play.

Value

The piece value of the huygens may depend significantly with the set of other pieces that are on the board, and their locations. The huygens can jump infinitely far, something even a queen cannot always do (a queen slides, so can be blocked by other pieces). But in most game setups, there are usually no pieces starting very far from the "a1-h8" area (the normal 8×8 chess playing area), and there's little reason for pieces to move far from this. Therefore the huygens gains little or no benefit from the infinite number of squares it attacks. With this in consideration, one study has shown that when played with other pieces starting in a zone of 18 files and 20 ranks, the huygens, with a shortest allowed jump of 5 squares, will typically have about nine usefull squares which it attacks, leading to an estimated value slightly superior to a bishop, or about 3.5 pawns.

Infinite Chess

Chess on an Infinite Plane starting position. Guards are on (1,1),(8,1),(1,8),(8,8). Hawks are on (-2,-6),(11,-6),(-2,15),(11,15). Chancellors are on (0,1),(9,1),(0,8),(9,8).

Infinite Chess is a term used to describe chess games with the variation that the board is unbounded. There's no indication of when such a chess-variant, or for that matter, any board game was first conceived. In chess-like games, perhaps ever since a player has had a king checked at the edge of the board has the concept of increasing the board size been imagined.

Classical chess, or FIDE-chess, is played on an 8x8 board with 64 squares total. Courier chess, played on a 12x8 board (96 squares), was played in the 12th century and continued to be played for at least six hundred years. There are also smaller versions of chess including Minichess, which has versions as small as 4x4. Magnus Carlsen has promoted a version of chess called "Chess Attack".

As for chess-like games larger than courier chess, there are plenty of examples, some going back centuries. One of the largest is Taikyoku shōgi (or "ultimate chess"). This is a chess-like game that was played in Japan in the mid 16th century and is played on a 36x36 board (1296 total squares). Each player starts with 402 pieces of 209 different types, and would probably take more than an hour to setup. The exact rules of the game are unknown and it is unclear if it was every widely played. But a well-played game would certainly require several days of continuous play, and require each player to make over a thousand moves.

Infinite Chess

Silverman 4×5 Chess

Taikyoku Shogi

Infinite chess represented by ASCII character symbols by Jianying Ji.

In the year 2000, a chess player by the name of Jianying Ji was one of many to propose infinite chess, suggesting the chess pieces remain in the same relative positions, but changing the board to an unbounded playing area. He represented the scheme using ASCII character symbols.

Numerous other chess players, chess theorists, and mathematicians who study game theory have conceived of infinite chess variations, often with different objectives in mind. Chess players sometime use the scheme simply to alter the strategy, requiring the players to find new checkmating patterns, which is necessary because there are no borders to trap the king. Theorists conceive of infinite chess variations expand the theory of chess in general, or as a model to study other mathematical, economic, or game-playing strategies.

One version of infinite chess played to day is Chess on an Infinite plane. Seventy-six pieces are played on an unbounded chessboard. The game uses orthodox chess pieces, plus guards, hawks, and chancellors. The absence of borders makes pieces effectively less powerful (as the king and other pieces cannot be trapped in corners), so the added material helps compensate for this.

Despite the infinite playing area, mathematical investigations have shown that in a general endgame, one player can force a win in a finite number of moves.

Other Variations

• Formation Chess, The Battle of Kadesh This is a variant where each player starts with fourteen knights and six other pieces. Knights can join into 2×2 groups gaining the ability to travel as a chariot force, gaining a queen's moves. Played on large or unbounded gameboards.

• Trappist-1 This is a variation which uses the huygens, possibly making the game mathematically undecidable.

See also

• Chess
• Chess variants
• Fairy chess pieces
• List of abstract strategy games
• Correspondence chess

External links

• Chess.com (Variant forum) Forum for discussing and playing chess variants.
• The Chess Variant Pages