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(2018-09-02)  
The Bible says Abraham  left Ur  to settle in theLand of Canaan.

The Game of Twenty Squares  has been known as:

• Illut Kalbi  (Pack of Dogs)  to the ancient Babylonians  (Akkadians).
• Asseb  or Aseb  to the Egyptians (mostly thestraightened version).
• Aasha  in the living memory of some older Cochin Jews whose community played it continuously until 1950 or so.

It's now named after the city of Ur, where the oldest extant boards  (c. 2600 BC) were excavated in the 1920's. More than one hundred more recent boards were found elsewhere.

The city of Ur  itselfwas founded around  3800 BC  in the Fertile Crescent (the purported cradle of civilization).

The ancient city of Ur was once the capital of Sumer and it rose to prominence again as capital of the Neo-Summerian Empire (Ur III, lasting for 115 years around 2000 BC) after the fall of the Akkadian Empire founded by Sargon, which had united Sumerian and Akkadian speakers under one rule (with widespreadbilingualism).

Ur is located near the confluence of the Tigris  andEuphrates, the two rivers which defined ancient Mesopotamia (etymologically, the land between the rivers). The two rivers join to form theShatt al-Arab waterway,  which marks the border between modern-day Iran  and Iraq and runs through a low plain for  200 km  before discharging intothe Persian Gulf. Ur was originally on the Euphrates,  but the river changed course during the fourthcentury BC and the city was abandonned.

On an expedition funded by the University of Pennsylvannia and the British Museum,SirLeonard Woolley (1880-1960) excavated five gaming boards in 1926-1927, including the iconic one on display at the British Museum.

Irving L. Finkel  (1951-)

Irving Finkel was hired by the Britsh Museum in 1979 as an expert on cuneiform,  the oldest type of writing. The  British Museum  has a collection of about 130000 cuneiform clay tablets. By far, the largest in the World.

In the early 1980s,  Finkel took a special interest in a cuneiform tablet excavated inIraq in 1880 and now known as  BM 33333B  (formerly identified as  Rm III, 6B). It's signed by Itti-Marduk-balalu  and dated 177 BC (possibly,  176 BC).

That tablet was intended for an audience who knew the Game of Twenty Squares  very welland it proposed new rules to rejuvenate the game and make it more interestingfor divination purposes.  From this,  Irving Finkel endeavored to reconstructwhat the basic rules really were at the time  (recall that the game had alreadybeen around for more than two millenia by then).

Finkel also relied on a photograph of a privately-owned tablet destroyed in WWI (which he identifies as DLB, after the name of its owner: Count Aymar de Liederkerke-Beaufort). The DLB tablet is undated but its script style indicates that it predatesthe aforementioned BM rablet by several centuries. Both tablets were discussed together in 1956 by the French Assyriologist Jean Bottéro(1914-2007). The DLB tablet makes it clear that it's primarily about a game (Akkadian: melultu)  and gives its ancient name: Illut Kalbi  (Pack of Dogs).

Then,  Finkel came across a photograph of a 20-square board belongingto a Jewish family from Cochin,  India. It seems that the game had been played continuously since ancient times in thatpart of the World until the 1950s,  when the community started to emmigrate to Israel.  Since Finkel's sister, Deborah Lionarons, was living in Jerusalem,  she went door-to-door witha picture of the board,  seeking older Cochin Jews  who might recognize it...

Ultimately,  Lionarons met Ruby Daniel, a retired schooteacher in her 70s,  who had left Cochin in 1951. As a child, she had played on paper layouts  (with 12 pieces,  instead of 5, 6 or 7) the game she called Aasha,  which matched closely what Finkelalready knew about the Royal Game of Ur.

(2018-09-06)  
There are two  single-lap variants.

When the Game of Ur  is played either for entertainment or gambling (as opposed to divination)  the markings on the squares are irrelevant exceptfor rosettes. There are two reasons why rosette  squares are desirable:

• Landing on a rosette gives you another move  (toss the dice again).
• You can't be dislodged from a rosette by an enemy piece  (it's safe).

The Game of Ur  is fundamentally based on the premises thatthe dice only allow a move of  4  squares of less. The game is thus arguably based on a regular design where every fourth square isa rosette.  This regularity holds for the normal rules  (regular lap  or long lap). Not for the short lap.

Basic Rules :

The following diagrams give the tracks followed by the pieces of the player whostarts and ends on the near side  (the other player uses a symmetrical track obtained by flipping horizontally the track of the near player).

After deciding  (possibly by tossing the dice)  which player goes first, the players take turns throwing the dice. After a toss,  the player advances one of his own pieces by the total numberof pips indicated by the dice,  according to the following constraints:

• A piece can't land on an already occupied rosette  square.
• A piece can't land on a square occupied by a piece of the same color.
• An exact count is required to bear a piece off the board.
• If a piece lands on a square occupied by an enemy piece, that piece is removed from the board  (it goes back to the starting position). This is called an attack.

A player must pass upon a zero toss or when there are no legal moves (otherwise,  the player must  move). A player who lands on a rosette plays again  (new toss). The winner is the first player whose pieces have all been born off (there are no ties in the Game of Ur).

	Short Lap   Bell's route  (1960) Last 2 squares are safe.
	Normal Lap Regular Route Long Lap

Egyptian Layout  (Straight Game of Twenty Squares) :

This last track uses a more recent type of board (first millenium BC).  It would be equivalent to the olderMesopotamian board with the convention that both players move clockwise after the bridge  (normally one playergoes clockwise and the other one goes counterclockwise).

The counterflow in the last section of the long-lap Mesopotamian layoutmakes enemy pieces easier to attack for a player who is substantially behind.

(2018-09-09)  
The second part of a circuit is performed with the piece upside-dowm.

Archeologically,  the coinlike pieces accompanying the Game of Ur gameboard have a quincunx  on one side, which makes some variants possible which involve flipping pieces. For example,  the circuit of every piece could consist of two consecutive simple laps.  The piece is flipped to indicate it's running its second and last lap.

Being at war  for the better part of two complete lapsmay be too much of a good thing,  though. The rules can be tuned in two different ways:

• Shortening the combined circuit of each piece.
• Disallowing some attacks based on the respective sides of the two pieces involved.

(2018-09-07)  
The Egyptian game  (1000 BC)  hade five different  pieces per player.

To prrperly analyze them,  it's best not to lump togetherthe two versions of the Game of Twenty Squares. Although the same games could certainly be played on both types of equipment, the distinction summarized by the following table does clarify things:

First Die	1	2	3	4
If successfully "doubled"	5	6	7	10

The birds  can enter the board only if thenumber of their home square is rolled.  They mustdo so in the order listed below, except for the eagle,  which can enter anytime the swallow is in play.

• 2 (1 token):  Swallow.
• 5 (2 tokens):  Seagull  (Babylonian storm-bird).
• 6 (2 tokens):  Raven.
• 7 (2 tokens):  Rooster.
• 10 (3 tokens):  Eagle.

(2018-09-03)  
Only the shape of the track varies from one version to the next.

The Royal Game of Ur  was played continuouslyfrom its creation to the early 1950s. Before its recent revival,  it was last played inthe Jewish community who had flourished in relative isolation within the Indian cityof Cochin until the creation of the State of Israel, where a substantial portion decided to immigrate.

Through its five millenia of active history, there's very little doubt that every possible variant of the game was played, especially considering how few of them are mathematically compatible with theprinciples which everybody  has always agreed on. Either players experimented on their own or they understood imperfectlythe rules they were first taught. All variants probably took root in some local communities at onetime or another,  with the possible exception of the versionswhich are called twisted  in the classification below (they impose a strategy so aggressive that the game tend to last very longwith an outcome which has very little to do with the skill of the player).

There are three independent ways the Royal Game of Ur  may vary:

• The number of pieces of each player.
• The way dice  are used.
• The shapes of the two symmetrical  along which the players move.

In this section,  we'll deal with the latest issue only.

The first lap  (either short or long)  always ends on a corner rosette. In single-lap variants,  pieces are simply born off after that point. (Purely for aesthetic reasons,  the final rosette used is in the player's side.)  Otherwise,  we flip the piece to indicate its second part is in progress.

If the trajectory is to proceed with a jump across the board's notch intoa launching pad  it makes a difference whether that rosette is on the player'sside  (untwisted)  or on the opposite side  (twisted). The second lap  (long or short)  is normally even  (ending on the player's side) but could also be odd  (ending on the opposing side).

Otherwise,  the trajectory backtracks into the central lane either directly or via a short or long loop. At the end if the central lane,  pieces are born off either directly (central)  through their own wing  (untwisted)  orthrough the opposing wing  (twisted).

1. Single short.
2. Single long.
3. Double short.
4. Double long.
5. Short-long.
6. Long-short.
7. Twisted short laps.
8. Twisted long laps.
9. Twisted short-long.
10. Twisted long-short.
11. Short backtrack.
12. Long backtrack.
13. Short central backtrack.
14. Long central backtrack.
15. Short twisted backtrack.
16. Long twisted backtrack.

In the twisted or switched variants,  the wing of a player isn't private.

(2018-08-31)  
Both players have  n  pieces  (usually,  n = 7).

The waydice are used is irrelevant to these enumerations of static positions.

Ancient coinlike pieces had a quincunx  on one side, which strongly suggests that flipping was involved,  in at least some variants of the game. For example,  a track to go through the center lane  (and possibly other squares)  in both directions, as in the rules concocted byDmitriy Skiryuk.  We won't consider that possibility here, which is unsupported by historical evidence.

Two natural symmetrical  tracks exists for the piecesof each player  (Black and White)  which keep the first four squares private:

• Short tracks :  14 squares,  8 shared ones  (central lane).
• Long tracks :  16 squares,  the last 12 ones are shared.

Short tracks of 14 squares with 8 shared squares :

The  n  pieces of either player can be found in the following locations:

• Off the board,  at departure  (all of them are there at first).
• Off the board,  at destination.
• On a private square  (the first four squares or the last two).
• On a square of the shared middle lane  (eight shared squares).

There can be at most one piece on any square of the board.

All squares have distinct positions along each 14-square track. In basic gameplay,  all the pieces of each player are alike.

The various square designs are ignored except for the rosettes at positions  4,  8  and  14  along both tracks. All rosettes give a free throw and the rosette at position  8, on the shared lane  is especially important because it marks the only safe square.

Let's first count the number of ways  p  pieces of either playercan be placed outside of the middle lane. We may put  q  pieces on the  6  private squares in one of C(6,q)  ways,  then the remaining p-q  pieces can be distributed between departure and destination in 1+p-q  ways.  The total number of distinct configurations is:

f (p)   =

(1+p-q)  C(6,q)

Now,  the number of configurations of the central lanecontaining  b  black pieces and  w  white pieces is the following multichoice number:

C(8,b,w)   =   C(8,b) C(8-b,w)   =   8! / [ b! w! (8-b-w)! ]

Therefore,  the total number of configurations with  n  pieces on each side is:

Ur (n)   =

C(8,b,w) f (n-b) f (n-w)

Thus,  with  7  identical pieces per player, the total number of diagrams  in the Royal Game of Ur is just under  141  million. Each such diagram corresponds to two positions (depending on whose turn it is to play).

Long tracks of 16 squares with 12 shared squares :

In the above text,  we may replace  C(6,q)  by  C(4,q)  to obtain:

We further replace  C(8,b,w)  by  C(12,b,w)  to obtain:

So,  with  7  pieces,  we have about  501  million diagrams for long tracks.

The Short Track is a Monstrosity

In the long track,  it always takes a jump of four squares to gofrom one rosette to the next. The same is true for a later version of the 20-square board with just two starting wingconsisting of a single standard section  (beginning with three regular square and ending with a rosette) and a straight central lane of three such sections. It's clearly meant to be played in only one way:  Down the central lane after the starting winguntil the piece goes back after being flipped on the final central rosette.

It's quite possible that the board was redesigned because ofthe ambiguity of the old layout was leading too many people astray.

We may call regular  a track made only from 4-square sections ending with a rosette.  The long lap is regular, the short lap isn't.

(2018-09-02)  
Three or four of these may be used in the Royal Game of Ur.

There is overwhelming archeological evidence that the game of Ur was playedeither with 2-sided stick dice or with special tetrahedral dice (sometimes improperly called pyramids). In modern parlance,  the latter variety,  on which we shall focus, are D2 dice. That's to say that each die is equally likely to produce one of twopossible outcomes  (1 or 0, marked or unmarked). Tossing an Ur die is just like flipping a fair coin.

Each die is a regular tetrahedron with two marked corners. When thrown,  a marked corner comes on top with  50%  probability.

When four dice are used,  the outcome of a throw is the number of uppermost marked corners. It's  0, 1, 2, 3, 4  with respectively  1, 4, 6, 4, 1  chances out of  16.

When three dice are used,  the same method is used except that the outcome is consideredto be  4  when none of the three top corners are marked. So,  the outcome is  1, 2, 3, 4  with respectively  3, 3, 1, 1  chances out of  8.

The odds are totally different.  So is the playing strategy. The reader is encouraged to check that the average jump on plain free track is exactly 2  square in either case.  However, from the initial position,  the player willhit the first rosette  (and get a second throw)  once in  16  times with four diceand once in  8  times with three dice.

In a pinch,  you may use coins instead of Ur dice.

(2018-09-02)  
Each entry will contain the probability of a win for White in that position.

The results of the above enumerations show that it'sentirely practical to work out a full Nalimov table for the entire Royal Game of Ur  to play it perfectly from the start. (there are only 141 or 501 million positions for the short or long laps,  respectively.)

As draws are impossible in the Game of Ur,  the probabilityof a win for Black is just the complement to  1  of the stated probability ofa white win.

To build a statistical Nalimov table, we first go though all the entries and determine how many descendants it has, for all possible throw of the dice.  That count is recorded within the table entry.

Then,  we work backward from every final position where the piecesof one of the players have all arrived and assign a value of zero or one to them. (We ignore the illegal position where the pieces of both  players are at destination.)

Everytime a final value is assigned to a position,  we decrement the count  of all its possible predecessors. When such a count reaches zero,  the corresponding is put into a stack, which contains the nodes whose values are ready to be computed.

Once an update is complete,  we process an element from the stack untilthe stack is empty.  At which point the whole Nalimov tablehas been computed.

If  1000  nodes are processed in one second,  about  86  millionwill be processed in a day and it takes only a couple of days to complete the computation.

To play with a pre-computed Nalimov table from a given position with a giventhrow of the dice,  we merely pick the available position with the best stored value.

(2018-09-04)  
The average rate at which a piece goes from departure to destination.

(2018-09-18)  
Under those rules,  we're no longer dealing with a race game !

The Ur Game™  now sold by Wood Expressions of Gardena, CA is Made in China. It's an imitation of the wooden classic originally designed and produced by Northwest Corner in1987.

The graphics are nice and the workmanship is flawless but the materials used aredefinitely on the flimsy side considering the hefty retail price I paid ($67 on Amazon; delivery took  16  days).  The engineering is minimal:

• Perpendicular cuts for square separators  (instead of 45° miters).
• Dead cavity beyond the bridge.
• Extremely  loose storage drawer. So much so that I intend to fit mine with a back-mounted magnetic latch.

The board features three distinct square designs which are repreated five times (the aforementioned rosette  is just one of those three). For the purpose of the following explanation,  we'll call the remaining squares unmarked  (actually, two pairs of identicalsquares and one unique one).

The object of this modern game is to occupy four identical squares. Both players have  6  pieces of opposite colors, each marked with a quincux one side. Initially,  all pieces show the quincux.

Phase 1:  Placement.

After using the dice to determine who goes first. The players take turn to place one of their pieces in any unoccupiedsquare they like.  during the game,  there's never morethan one piece in a square.

A player may secure an easy win  in this phaseby placing four pieces on identically-marked squares. This can only happen if the opponent blunders the game away.

Phase 2:  Movement.

Each player in turn moves on of his piecesby the number of squares determined by the dice, either horizontally,  vertically or diagonally, into an unoccupied square. The piece so moved is flipped and cannot move again untilall six pieces show the same side.

The number given by the dice is counted as the number of different  unoccupied squares on a path from originto destination through a sequence of squares which are adjacentlaterally of diagonally.  Such a path could possibly cross or retrace itselfbut only free squares are counted  (and each free square is only counted once). The terse Northwest rules  are ambiguous on that last pointbecause they also mention that a player could also win if theopponent is unable to move  (which is not possible with the aboveway of counting).