"Board and Table Games from Many Civilizations" by R. C. Bell (1960, 1969).
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 Abraham 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).
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 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 Christ.
Cuneiform 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and most of the one to two million extant tablets remain unstudied and unpublished.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 (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 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 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.
(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).
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 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) hade five different pieces per player.
To prrperly 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:
Qualifier
Name
Board Layout
Pieces
Date
Mesopotamian
Game of Ur
Wings, aisle and bridge to island
Identical (double-sided)
2600 BC
Egyptian
Asseb
Wings and long aisle
5 different birds
1000 BC
First Die
1
2
3
4
If successfully "doubled"
5
6
7
10
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.
(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 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).
Single short.
Single long.
Double short.
Double long.
Short-long.
Long-short.
Twisted short laps.
Twisted long laps.
Twisted short-long.
Twisted long-short.
Short backtrack.
Long backtrack.
Short central backtrack.
Long central backtrack.
Short twisted backtrack.
Long twisted backtrack.
In the twisted or switched variants, the wing of a player isn't private.
(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) =
p
å
q = 0
(1+p-q) C(6,q)
Number of configurations of p pieces outside the 8 shared squares.
p
0
1
2
3
4
5
6
7
8
9
10
p ≥ 5
f (p)
1
8
30
72
129
192
256
320
384
448
512
64 (p-2)
Now, the number of configurations of the central lane
containing b black pieces and w white pieces is the
following multichoice number:
Therefore, the total number of configurations with n pieces on each side is:
Ur (n) =
n
å
b = 0
n
å
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
0
1
2
3
4
5
6
7
Ur (n)
1
248
13112
272641
2486427
15788044
55892884
140939686
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
0
1
2
3
4
5
6
7
8
9
10
p ≥ 3
f (p)
1
6
17
32
48
64
80
96
112
128
144
16 (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
0
1
2
3
4
5
6
7
Ur (n)
1
312
20623
514052
6165948
41264288
172726782
501032952
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.
(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.
(2018-09-04) Best strategy with many pieces.
The average rate at which a piece goes from departure to destination.
(2018-09-18) 2D modern game with Mesopotamian board & pieces:
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
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 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 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 pieces 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).
