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Final Answers
© 2000-2018   Gérard P. Michon, Ph.D.

Royal  Game  of  Ur
( Game of Twenty Squares )

Backgammon is a better version of this.
Irving Finkel   (1951-) 

Also on this site:  Royal Game of Ur

Related Links (Outside this Site)

On the Rules for the Royal Game of Ur  by  Irving L. Finkel  (2007-04-20).
20-Squares:  The Ancient Board Game  by  John Bardinelli.
The Rules of the Royal Game of Ur  (Masters Traditional Games).
Royal Game of Ur  by  Catherine Soubeyrand   (Game Cabinet).
Senet and the Royal Game of Ur  by  Jenny Williams  (2010-09-14).
The Royal Game of Ur  by  Boards and Pieces.
Game of Ur:  A blueprint for better games?   Namir Ahmed  (2013-01-15).
Twenty Squares  by  Anne-Elizabeth Dunn-Vaturi  (The Met, 2014-12-09).
Adventures with the Royal Game of Ur  (The Getty Villa, 2017-04-02).
Royal Game of Ur  and  Game of 20 Squares  (Cyningstan).
Royal Game of Ur  and  Aseb  by  Eli  (Ancient Games, 2017-09-19).
Watch the world's oldest board game being played  (2018-04-05).
Play Ur online  (single short lap)   |   Royal Game of Ur  (Wikipedia).

History of Writing Systems  (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) NativLang  (2015).
Cuneiform hand-me-downs:  How Sumerian outlived its speakers (2:49)  by  Joshua Rudder  (NativLang, 2015-04-24).
Deciphering the world's oldest rule book (7:02)  by  Irving Finkel  (2015-09-23).
Tom Scott vs. Irving Finkel (25:32)  The British Museum  (2017-04-28).
Complex rules in 177 BC (16:56)  Irving Finkel  (Geek & Sundry, 2017-05-12).
Review and Tutorial (20:06)  The Board Game Kaptain  (2017-06-27).
Game Play Variations (13:20)  3D Printing Professor  (2017-06-27).
Tabletop Simulator (15:11)  Steve and Stephan  (Freak Occurence, 2017-11-02).
Royal Game of UR:  AI on Sharp EL-5500II (34:47)  Tinker Life  (2017-12-21).
"Board and Table Games from Many Civilizations"  by  R. C. Bell  (1960, 1969).

 Royal Game of Ur

Analyzing  the  Royal Game of Ur

(2018-09-02)   History of the  Royal Game of Ur.
The Bible says  Abraham  left  Ur  to settle in the Land of Canaan.

Although the city discussed below is clearly what the Bible calls  Ur of the Chaldees  (Ur Kasdim)  some scholars believe the birthpkace of Abrabraham to be some  lesser-known place  with a similar name which biblical scribes originally mistook it for,  possibly the rural land of  Ura,  near the city of  Harran,  in  Upper Mesopotamia.

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.
  • 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  itself was 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 widespread bilingualism).

The  Ur-III  period saw a revival of the old  Sumerian language  for religious and official purposes  (in much the same way Latin was used in Europe in the Middle Ages, Renaissance and beyond)  but  Akkadian  remained the common language,  eventually morphing into  Aramaic,  the semitic  lingua franca  spoken by  Jesus ChristCuneiform  writing was originally invented for the ancient Sumerian language,  but it was adapted to convey all languages of the region as long as clay tablets were in use.  The script itself became extinct around the second century AD and had to be deciphered from scratch in the 19th century.  Today,  there are only a few hundred  cuneformist  and most of the one to two million extant tablets remain unstudied and unpublished.

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

The nearby city of  Eridu  is also a major archeological site/  It's the oldest city in the World:  Eridu was founded around  5400 BC.

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

 Irving L. Finkel

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 in Iraq 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 well and it proposed new rules to rejuvenate the game and make it more interesting for divination purposes.  From this,  Irving Finkel endeavored to reconstruct what the basic rules really were at the time  (recall that the game had already been 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 predates the aforementioned BM rablet by several centruries.  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 belonging to a Jewish family from Cochin,  India.  It seems that the game had been played continuously since ancient times in that part 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 with a 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 Finkel already knew about the  Royal Game of Ur.

Royal Cemetery at Ur   |   Royal Tombs of Ur
Treasures from the Royal Tombs of Ur (12:34)  Saint Louis Art Museum  (2011-04-25)
2018 Sumerian Documentary:  Secrets of Ur (44:46)  Legends of Mesopotamia  (2018-04-11)
Nouvelles de Mésopotamie (3:33 in French)  by  Jean Bottéro  (Musée du Louvre,  1996).
Big Game Hunter  by  William Green  (Time, London, 2008-06-19).
Traditional board games:  From Kochi to Iraq  by  Priyadershini S.  (The Hindu,  2015-10-01).
A Game Comes Alive  by  Priyadershini S.  (The Hindu,  2015-10-02).
The Indiana Jonesian discovery of the Royal Game of Ur  (Curio Games, 2016-06-14).
The Royal Game of Ur  World History  (BBC, 2014).

(2018-09-06)   Simple rules for the  Royal Game of Ur.
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 except for  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 that the dice only allow a move of  4  squares of less.  The game is thus arguably based on a regular design where every fourth square is a 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 who starts 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 number of 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).

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

Egyptian Layout  (Straight Game of Twenty Squares) :

 Royal Game of Ur 
 (Egyptian Layout)

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

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

(2018-09-09)   Complex Circuits
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 laps may 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)   Mesopotamian and Egyptian pieces:
The Egyptian game  (1000 BC)  could accomodate five  different  pieces per player.

To porperly analyze them,  it's best not to lump together the 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:

QualifierNameBoard LayoutPiecesDate
of Ur
Wings, aisle and
bridge to island
2600 BC
EgyptianAssebWings and long aisle5 different birds1000 BC

First Die  1    2    3    4  
  If successfully "doubled"  56710

The  birds  can enter the board only if the number of their home square is rolled.  They must do 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.

 Come back later, we're
 still working on this one...

Harvard Semitic Museum
Simple Rules for the Game of 20 Squares (4:24)  Adam J. Aja & Robyn Klaus.
Complete Rules for the Game of 20 Squares (13:44)  Adam J. Aja & Robyn Klaus.
History of Ancient Egypt - 8000 BC to 30 BC  (1:59:56)  by  Justin Walsh  (2013-09-16).

(2018-09-03)   All Variants of the  Royal Game of Ur.
Only the shape of the track varies from one version to the next.

The  Royal Game of Ur  was played continuously from its creation to the early 1950s.  Before its recent revival,  it was last played in the  Jewish community  who had flourished in relative isolation within the Indian city of  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 the principles which  everybody  has always agreed on.  Either players experimented on their own or they understood imperfectly the rules they were first taught.  All variants probably took root in some local communities at one time or another,  with the possible exception of the versions which are called  twisted  in the classification below  (they impose a strategy so aggressive that the game tend to last very long with 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 into a launching pad  it makes a difference whether that rosette is on the player's side  (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)  or through 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.

 Come back later, we're
 still working on this one...

(2018-08-31)   Counting the diagrams and positions in the  Game of Ur.
Both players have  n  pieces  (usually,  n = 7).

The way dice 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 by Dmitriy Skiryuk.  We won't consider that possibility here,  which is unsupported by historical evidence.

Two natural  symmetrical  tracks exists for the pieces of 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.

In either case,  there are two  mathematically equivalent  possibilities,  if we only assume that the tracks are continous and symmetrical to each other.  We retain the one where pieces of a given color start and end on the same side of the board  (at the same notch).  This standardization also suggest a very simple two-lap variant,  which was probably used in antiquity when a longer game was desired,  namely:  A piece must go around the track  (short or long)  twice  before reaching the destination and it's flipped when it crosses the board's notch,  to indicate that it's going around for the second  (and last)  time.

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 player can 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)   =   
q = 0
   (1+p-q)  C(6,q)

Number of configurations of  p  pieces outside the  8  shared squares.
p 012345678910p ≥ 5
  f (p)   18307212919225632038444851264 (p-2)

Now,  the number of configurations of the central lane containing  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)   =   
b = 0
w = 0
    C(8,b,w)  f (n-b)  f (n-w)

Number of configurations with  n  pieces per player for  8  shared squares.
n 01234567
  Ur (n)   12481311227264124864271578804455892884140939686

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:

Number of configurations of  p  pieces outside the  12  shared squares.
p 012345678910p ≥ 3
  f (p)   1617324864809611212814416 (p-1)

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

Number of configurations with  n  pieces per player for  12  shared squares.
n 01234567
  Ur (n)   131220623514052616594841264288172726782501032952

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 go from one rosette to the next.  The same is true for a later version of the 20-square board with just two starting wing consisting 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 wing until the piece goes back after being flipped on the final central rosette.

It's quite possible that the board was redesigned because of the 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)   The Ur binary dice.
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 played either 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 two possible outcomes  (1 or 0, marked or unmarked).  Tossing an Ur die is just like flipping a fair coin.

On the other hand, traditional two-sided stick dice are rounded on one side and flat on the other.  This asymmetrical design doesn't guarantee them to be  fair.

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 considered to 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 will hit the first rosette  (and get a second throw)  once in  16  times with four dice and once in  8  times with three dice.

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

If you absolutely must use a six-sided die for the sake of convenience,  the least damaging way to do so is to interpret 5 as zero and 6 as two.  This does give an average of  2  and makes that average more likely than the other outcomes,  as it should.

How to customize blank D4 Dice for Royal Game of UR (5:25)    (2017-10-03).

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

The results of the above enumerations show that it's entirely 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.)

The two-lap variants are not amenable to that brute-force approach,  since their total numbers of positions are  much  larger.

As draws are impossible in the  Game of Ur,  the probability of a win for Black is just the complement to  1  of the stated probability of a 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 pieces of 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 until the stack is empty.  At which point the whole Nalimov table has been computed.

If  1000  nodes are processed in one second,  about  86  million will 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 given throw of the dice,  we merely pick the available position with the best stored value.

 Come back later, we're
 still working on this one...

(2018-09-04)   Best strategy with many pieces.
The average rate at which a piece goes from departure to destination.

 Come back later, we're
 still working on this one...

(2018-09-18)   2D modern game with Mesopotamian board & pieces:
Under those rules,  we're no longer dealing with a  race game !

 Northwest Corner

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  in 1987.

On  1991-03-15,  the  British Museum  acquired a copy of the original  Northwest edition  (so-called)  possibly through curator  Irving Finkel.

The graphics are nice and the workmanship is flawless but the materials used are definitely 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 features three distinctive 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 thet feature two pairs of identical squares 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 piece in any unoccupied square they like.  during the game,  there's never more than one piece in a square.

A player may secure an  easy win  in this phase by 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 pieces by 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 until all 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 origin to destination through a sequence of squares which are adjacent laterally of diagonally.  Such a path could possibly cross or retrace itself but only free squares are counted  (and each free square is only counted once).  The terse  Northwest rules  are ambiguous on that last point because they also mention that a player could also win if the opponent is unable to move  (which is not possible with the above way of counting).

 Come back later, we're
 still working on this one...

visits since September 2, 2018
 (c) Copyright 2000-2018, Gerard P. Michon, Ph.D.