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The Mathematical Board 

". . . entonces verán que era su sabiduría la que tenía poder sobre sus vasallos 
y solemnemente les será devuelta su estera, símbolo del poder . . ." 

Chilam Balam de Chumayel. 

Lost in the penumbra of times, the origin of the board, utilized in chess and 
checkers, is mysteriously mixed with the development of the mathematical science. 

The diverse legends and myths about this invention are associated invariably to 
people or cultures which were acknowledged to have an advanced knowledge of 
numerals. Thus, the Romans received it from the Greeks, the Greeks from the 
Egyptians and the Babylonians learned it from the Indostans. While the Jews 
attributed the merit to Solomon - the most wise man among their kings - the 
Greeks assign it to Hermes, the Chinese to the Mandarin Hansing, the Hindus to 
Brahman Sissa and the Persians to the connotated astronomer Shatrencha. A major 
number of particulars, however, point to the fact that the board as well as 
chess and checkers games existed in India and there is mention of documents with 
previous dates. From there it seems to have migrated to Persia at the beginnings 
of the Christian Era and distributed through Europe from Constantinopla to the 
Spain occupied by the Moors. 

If at the beginning the board had a mathematical function, later it was 
converted to a mysterious instrument of magic rituals and fortunespelling and 
end as a simple hobby. The science of one day becomes the magic of the next day 
and diversion in another time. Thus, there was a time when the hidden secret of 
the scientific character of the board was diluted by esoterism of the Kaballah 
and proliferation of magic squares where the numbers were combined in ingenious 
modes and symbolisms. The masonic ritual has preserved until our times a varied 
range of symbols inherited from an archaic cultural substratum. Examples are the 
floor with white and black mosaics, the numerals of the sacred stones, the 
masonic alphabet derived from a square box with dots. 

Further, the board came to represent the mathematical plot of the universe on 
which the human knowledge is placed. For this reason, it appears on floors of 
masonic and rosicrucian temples as well as in the reticle that covers the 
internal walls of the Quadrangle of Las Monjas and the Governor's House in Uxmal. 
Also, it represents the crest of Mayan temples and observatories as an 
architectonic element through several thousands years of civilization. 

In the garments of the Inca dignataries, Guamán Poma de Ayala draws squares with 
inscribed numbers. The reader can compare his drawings with the post-hispanic "ponchos" 
to appreciate the reticulated design and discover some signs used by the Mayas 
with a mathematical meaning. 

The stelas of Yaaxchilán, Piedras Negras, Copán y Quirigua show mantles and 
flaps of Maya priests with designs including squares and inscribed numbers. In 
Palenque exists several altarpieces dedicated to persons of high hierarchy where 
the most relevant feature is a shawl with squared designs and dots in the center 
of each square. This fact suggests the recognition of a title or rank where the 
mathematical knowledge is preponderant. 

For the common calculations, it is probably that a board with nine squares was 
enough. An example is the magical table of Cornelio Agripa that was consecrated 
to the god Saturn or Chronos who is also associated with calendar calculations. 
It is useful to remember that Saturnian gatherings were events where the Greeks 
and Romans allow random or chance games and are tightly related with the 
mathematical techniques that I am investigating. A link is put into evidence: 
the probabilities are a branch of the mathematical knowledge. 

In all symbolic boards columns and rows are very important as well as diagonals. 
As we see later, it obeys an important function contained in the three elements 
for the manipulation of tokens or chips. 

It is not a surprise that the Mayas used boards to make their calculations, 
however, it arises the question of why these wooden or stone boards are not 
found in the Mayan zone as it happens in Perú. It exists in some mysterious 
squared designs on the floors of the Palace of Zacuala, in Teotihuacán, in the 
Chamber (closed) of the pyramid of Uxmal, in the castle of Chichén, and probably 
in other places but not in enough quantity to bring the attention of 
archaeologists. There are some explanations to the point. The floors have 
experienced an intense wear and tear by traffic and any painted square or line 
on them has the probability of have being raised. It could also be thought of 
the fact of poor durability of wood in a wet climate and did not have the 
advantage of being placed on doorways such as Tikal. We need to add that no one 
has been concerned in looking for boards on the floors, and probably now we will 
have more frequent reports of such findings. 

Nonetheless, there is another reason that I wish to present as a hypothesis. One 
of the most solid antecedents in the use of boards is the placement of grains as 
tokens in a game known as "patolli," whose main players were the Aztecs. Says 
Mena and Jenkins (1) that ". . . Over a mat, e.g., rug made with hemp, a square 
is painted and it is crossed with two doubled diagonals, and in the point of 
intersection a square is centered and divided by for equal parts. Each one of 
the X-shaped cross is divided in twelve boxes. Near the extremities of each arm, 
within a wider box than the others a sign Nahui Ollin is painted . . ." 

". . . The players are seated in small chairs, one in between every two arms as 
they throw drilled grains, shaken with their hands, before they are thrown. The 
grains must fall in the boxes. If they fall outside the boxes, the game is lost. 
As noticed, the number of total boxes is 52, 48 boxes in each arm and 4 boxes in 
the center, that are the years contained in the Mayan cycle. The number of plays 
must match the astronomical cycle calculation. The game is absolutely 
astronomical, in a literal sense." 

More over, the same authors inform us that " . . . the players of patolli 
visited some Mayan festivals with a rolled mat under their arms and perforated 
grains linked by a thread . . ." 

The codexes Durán and Magliabechi show explanatory drawings of the patolli game. 
Attention is called to the numerous coincidences with the Mayan techniques of 
calculations. Then, we get to the conclusion that the patolli game, chess and 
checkers were originated from the mathematical calculations as it could happen 
in the near future we develop intelligence games supported by pocket electronic 
brains. 

The Florentine Codex (2) shows a drawing where Cipactonal and Oxomoco, the 
couple of Aztecs cosmic progenitors who have been identified with Ixmucané and 
Ixpiyacoc of the Mayas-Quichés, invent the calendar. While Oxomoco appears 
manipulating some corn grains, Cipactonal has in his hands a bow with knots, 
that is a quipu. The significative detail is that Oxomoco is not throwing the 
grains on the ground but over a mat. It seems logical to assume that the 
mathematical boards of Mayas were mats. It has always been an intrigue that a 
mat, "póop" in maya, plays a role so prominent in their symbolism and considered 
as synonym of the throne. In fact, "to sit over the mat and in power," as 
expressed in the books of Chilam Balam, is the act of taking the command and 
government of Mayan people, and, in occassions, is given the rank of cosmic 
domain by suprahuman entities. The first month of the Maya civil year is called 
Poop. How do we compare a mat with such important functions? 

It is evident that the mat contains a symbolism of great importance and it was 
believed to be found in the woven fabric that forms a repetition of crossed 
bands denominated "nahui ollín." This symbol, an admirable synthesis of the 
philosophical thinking of Mayas, occupies a preeminent place in the iconography 
of Tajín and in the Altiplano cultures. Therefore, it is not an speculation to 
think that the Mayan mats had a meaning directly related with the complex web of 
their society and dynamic of the Universe. 

But now, another fact emerges: the possible use of the mat to paint over the 
surface squares with a function for mathematical calculations. Before this 
possibility, the content of this symbol integrates fully government and power. 
Not only to show the tight interrelation of the four elements or Universe 
appearances to form a microcosmic and macrocosmic weave but representing the 
intellectual power of Man, able to computing, anticipation and planning. 

For these reasons, I forward the opinion that the mathematical boards were, to 
great extent, mats which are not existent due to the lack of discovered samples. 

With the purpose of establishing clear references, I will use a Java programmed 
board similar to the illustration shown below. 

The columns are identified by arabic numbers and the rows or horizontal lines by 
letters. The start point of the numbers and letters are in the top left corner. 
In reality we can arbitrarily choose any of the four corners but it is logical 
to be inclined more toward that corner because all Mayan inscriptions coincide 
this point with the beginning. 

We call the diagonal that starts from the top left corner and ends at the bottom 
right corner, the main diagonal. On the other hand, we designate the diagonal 
that begins at the top right corner and ends at the bottom left corner, the 
secondary diagonal. Each square is identified with a row number and column 
number (e.g. A for the row and 1 for the column) that belongs to. 

The operations of calculus in the Maya system are manual. It results very 
difficult to describe them with words and diagrams because it requires an effort 
of imagination to assimilate them mentally. However, if they are executed with 
the help of graphical interfaces such as the one I am proposing in this Web site 
these operations are understood quickly. However, the reader can learn them with 
a paper sheet, beans and half toothpicks. 

With these previous explanations, this project starts as we prepare to enter in 
a fascinating world of Mayan computers and the beauty of a Mayan universe never 
shown before. Enjoy it! 

References 

1. Mena, R. and Jenkins, A. Juan. Educación Intelectual y Física entre los 
Nahuas y Mayas Precolumbinos. México. pp. 37, 38, 44. 

2. Códice Florentino. Fray Bernardino de Sahagún. Santa Fe. Nuevo México. 1950-52. 

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