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In the quipu recording devices of the Andean peoples, knots were used to indicate numeric values based on their positions and types. Similarly, the positions of each of the squares on a yupana board indicated numeric value. Poma de Ayala's yupana board consists of five groups of four squares in which groups indicate ten thousands, thousands, hundreds, tens and ones, respectively. Within each group, there are four squares that represent the values 1, 2, 3 and 5.

The value being represented on a yupana is encoded by placing zero, one or more markers on the squares. Pereyra (1990) and Prem (2019) suggest the following arrangement of markers for the integers between 0 and 9:

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,

,

,

,

,

,

,

,

,

Using these arrangements, any non-negative integer (as long as it fits within the number of available columns) can be represented on the yupana, and by using differently colored markers, more than one number can be represented at the same time.

Summation of multiple numbers can be performed through the mechanical operations of ungrouping, moving and minimizing the number of markers on the yupana. In particular, these operations can be iterated until a ending state is reached in which no further actions can be made. At this point, the sum can be read directly off the ending configuration of markers. This process can be viewed in snapshot form using the following controls:

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Historical orientation Ending state (solution) Red beans 2020 Green beans 18085

Computational Explanation

The yupana, whose name derives from the word yupay—meaning to count in the indigenous Andean Quechua language—was the ancient calculating device of the Incas. Several historical accounts mention the existence of this device and describe it as a sort of tablet upon which stones or beans were placed. These tokens were manipulated in order to perform the various calculations which could be recorded on knotted strings known as quipus:

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⟶

Both the quipu and the yupana are shown in a 1615 drawing from the 1,200-page manuscript New Chronicle and Good Government written by Quechua nobleman Felipe Guaman Poma de Ayala. In the illustration, the quipucamoyoc (chief accountant and treasurer) is shown holding a quipu while a yupana board is depicted in the lower-left corner:

Quipus were used across the Incan Empire for storing and transferring exact numerical data. To perform arithmetical computations, quipucamayocs transferred numbers encoded by knots into the yupana using markers (typically beans, corn seeds, tokens or pebbles). The yupana enabled them to perform large computations effortlessly following a set of simple rules consisting of moving and adding or subtracting these markers until a ending state was reached:

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Computing Large Sums

In quipus used for record keeping, the types of knots and their positions indicated numeric value. Similarly, the positions of the squares on a yupana board indicated their numeric value. The yupana board in Poma de Ayala's drawing has five groups of four squares each, indicating ten thousands (hunu), thousands (waranqa), hundreds (pachaq), tens (chunka) and ones (-yoq/-niyoq), respectively:

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Each grouping has four squares with 1, 2, 3 or 5 dots engraved on them:

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Interestingly, these values correspond to the Fibonacci numbers 1, 2, 3 and 5:

In[]:=

Fibonacci[Range[2,5]]

Out[]=

{1,2,3,5}

Numbers on a yupana are indicated by placing markers on the squares, where the absence of markers indicates the number zero:

The appearance of Fibonacci numbers means that the representations of 2, 3 and 5 are all made by combining the values of the two previous squares:

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+



,

+



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+



In other words, since:

In[]:=

Fibonacci[Range[5]]

Out[]=

{1,1,2,3,5}

… the sums of pairs of consecutive Fibonacci numbers give:

In[]:=

Most@Total[Partition[Fibonacci[Range[5]],2,1],{2}]

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{2,3,5}

For the numbers 4, 6, 7, 8 and 9, more than one marker is needed. Pereyra (1990) and Prem (2019) suggest the following arrangement of markers:

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,

,

,

,

The ubiquitous modern algorithm for adding large numbers with pen and paper is to align the addends vertically and add the columns starting from the ones column on the right. If a column exceeds 9, the extra digit is carried into the next column. For example:

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Use long addition: 1 8 0 8 5 + 2 0 2 0

5+05: 1 8 0 8 5 + 2 0 2 0 5

Add: 8+210. Carry the tens place to the next column: 1 1 8 0 8 5 + 2 0 2 0 0 5

1+0+01: 1 1 8 0 8 5 + 2 0 2 0 1 0 5

Add: 8+210. Carry the tens place to the next column: 1 1 1 8 0 8 5 + 2 0 2 0 0 1 0 5

1+12: Answer: 1 1 1 8 0 8 5 + 2 0 2 0 2 0 1 0 5

In contrast, the yupana does not require carries or the computational steps to be followed in any particular order. In fact, learning and then applying a set of five basic moves (cf. Prem 2019) in any order is sufficient to obtain a correct summation:

Kikin ("the equivalent ones" in Quechua) move; corresponding to

1+12

:

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

Iskay ("two" in Quechua) move, corresponding to

2+24

:

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

Kimsa ("three" in Quechua) move, corresponding to

3+36

:

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

Pisqa ("five" in Quechua) move, corresponding to

5+510

:

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

Pichana ("cleaning brush" in Quechua) move, corresponding to

1+23

and

2+35

:

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

and



For example, if red and green beans are used as markers for 2,020 and 18,085, respectively:

… the addition of 2,020 + 18,085 is reduced to five basic movements on the yupana:

A pair of pichana moves:

A pair of pisqa moves:

And a kikin move:

Therefore, the total sum obtained on the yupana is 20,105:

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Notice that there is no need to follow a particular order since all strategies converge to the same ending state. As a result of the fact that this method of addition feels more like playing a board game than performing traditional and somewhat laborious pen-and-paper arithmetical computations, it potentially provides a fun way in which addition could be taught in elementary mathematics classes.

A starting kit to explore and learn about addition and other arithmetical operations on the yupana known as Yupana Inka Tawa Pukllay has been created by a Peruvian team lead by the systems engineer Dhavit Prem:

Historical Accounts

The first attempt to understand how the yupana worked was made by Henry Wassén in 1931 in an article entitled "The Ancient Peruvian Abacus," which appeared in a collection of comparative ethnographical studies edited by Erland Nordenskiöld. Like most recent interpretations, Wassén's interpretation is based on two written accounts of Incan arithmetic. The first of these accounts is the manuscript accompanying Guaman Poma de Ayala's drawing:

"Haciendan Chasquicoc, the main treasurer, says that Tahuantisuyo Runa Quipoc Incap, the main accountant, had great ability. In order to determine his skill, the Inca sent him to count and adjust Indian accounts in this region. . . His skill was greater than it would have been with pen and paper. [Quipucamoyocs] count with boards. They number 100,000 and 10,000 and 100 and 10 until arriving at 1. They count everything that goes on in the kingdom: the parties, the Sundays, the months, the years, and these eight treasurers and accountants were in every city, village, and Indian town of the kingdom. They counted in this way: starting with one, two and three- 1 – 2 – 3 – 4 – 5 – 6 – 7 – 8 – 9 – 10 – 20 – 30 – 40 – 50 – 60 – 70 – 80 – 90 – 100 – 1000 – 1000 – 10000 – 100000 – 1000000 – 10000000 – infinity."

The second reference used is that of José de Acosta taken from La Historia Natural y Moral de Las Indias, originally published in 1586:

"To see them use another kind of quipu with corn seed is a perfect joy. In order to effect a very difficult computation for which an able calculator would require pen and ink for the various methods of calculation these Indians make use of their seeds. They place one here, three somewhere else and eight I do not know where. They move one kernel here and three there and the fact is that they are able to complete their computation without making the smallest mistake. As a matter of fact, they are better at calculating what each one is due to pay or give than we should be with pen and ink. Whether this is not ingenious and whether these people are wild animals let those judge who will! What I consider as certain is that in what they undertake to do they are superior to us."

The Quinario-Decimal Structure of the Aymara Language

In an article entitled "Los Sistemas Numericos Del Quechua y el Aymara," Andean language expert Guido Alfredo Pilares Casas noted a connection between the yupana and the structure of the ancient Andean language of Aymara (see also Leonard and Shakiban 2010). Interestingly, this language is still spoken in South American several countries, particularly Bolivia and Peru:

In[]:=

LanguageData

Central Aymara

LANGUAGE

,"CountryLanguageFractions"

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

Bolivia

0.681647,

Peru

0.302575,

Chile

0.0129773,

Argentina

0.00280036

Historically, Aymara was endemic to the region of Potosí, Bolivia, which is the location in which the yupana pictured here was found:

Casas (2005) has considered the distribution of dots into groups of 1, 2, 3 and 5 from a linguistic perspective. As noted by Leonard and Shakiban (2010), it has been suggested that Poma de Ayala, the chronicler and author of the yupana drawing, spoke a form of Aymara. An interesting aspect of Aymara is that it uses a quinario-decimal system in which 5 is used in addition to 10 when forming number words, so instead of 27 being spoken as "twenty-seven" (as it would in a purely decimal number system), in Aymara, it would instead be formed as something like "twenty-five-and-two."

To see the significance of this fact, note that the three numbers incrementally larger than 5 (called qallqu in Aymara) on the yupana can all be derived by adding 1, 2 and 3 to the number 5 to obtain 6 (maqallqu = ma + qallqu = 1 + 5), 7 (paqallqu = pa + qallqu  2 + 5) and 8 (kimsaqallqu = kimsa + qallqu  3 + 5):

The numbers in the Aymara language therefore can be seen to coincide with the structure of yupana, making it especially accessible to Andean people speaking Aymara. As a result, the Yupana is used today as part of the curriculum in the rural, bilingual schools of Peru and Bolivia.

Other Resources

Additional Reading

• Acosta, J. La Historia Natural y Moral de Las Indias. 1586. Reprinted Mexico: Fondo de Cultura Economica, 1940.
• Apaza Luque, H. J. "La yupana, material manipulativo para la educación matemática." Ph.D. thesis. Madrid, Spain: Universidad Autónoma de Madrid, 2017. [link]
• Ascher, M. "Mathematical Ideas of the Incas." Ch. 10 in Native American Mathematics (Ed. M. P. Closs). Austin, TX: University of Texas Press, pp. 261–289, esp. pp. 264–266, 1996.
• Burns Glynn, W. "La tabla de cálculo de los Incas." Boletín de Lima, Vol. 11, pp. 57–70, 1981.
• Casas, G. P. "Los Sistemas Numericos Del Quechua y el Aymara." Revista andina, Vol. 40, pp. 149–178, 2005.
• Day, C. Quipus and Witches' Knots: The Role of the Knot in Primitive and Ancient Cultures. Lawrence: University of Kansas Press, 1967.
• Depaulis, T. "Inca Dice and Board Games." Board Games Studies, Vol. 1/1998, pp. 26–49, 1998.
• Leonard, M. and Shakiban, C. "The Incan Abacus: A Curious Counting Device." Journal of Mathematics and Culture, Vol. 5, pp. 81–106, 2010. [link]
• Locke, L. L. "The Ancient Peruvian Abacus." Scripta Mathematica, Vol. 1, pp. 36–43, 1932.
• Pareja, D. "Instrumentos Prehispanicos de Calculo: el Quipu y la Yupana." Integracion, Vol. 4, pp. 37–56, 1986.
• Pereyra, H. "El Antiguo ábaco peruano según el manuscrito de Guaman Poma." In Quipu y Yupana. Consejo Nacional de Ciencia y Tecnología. Lima, Perú, 1990.
• Poma de Ayala. G. F. The First New Chronicle and Good Government: On the History of the World and the Incas up to 1615 (Trans. and ed. R. Hamilton). Austin, TX: University of Texas Press, 2009.
• Prem, D. Yupana Inka: Decoding the Inka's Math. English Edition Kindle, 2019. [link]
• Tun, M. "Yupana." In Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (Ed. H. Selin). Dordrecht, Netherlands: Springer Science+Business Media, 2014. [link]
• Valera, B. Exsul Immeritus Blas Valera Populo Suo e Historia et Rudimenta Linguae Piruanorum Indios. Napoli: Archivio Miccinelli-Cera, 1618.
• Wassén, H. "The Ancient Peruvian Abacus." In Origin of the Indian Civilizations in South America (Ed. E. Nordenskiold). Goteborg, Sweden: Elanders Boktryckeri Aktiebolag, 1931.

Additional Links

• Dumbarton Oaks Research Library and Collection: The Role of Knots
• The Guaman Poma Website: A Digital Research Center of the Royal Library, Copenhagen, Denmark
• MALI, Museo de Arte de Lima: Yupana—Inca style 1400 AD - 1532 AD
• Metropolitan Museum of Art: Game Board (yupana)
• Museo Chileno de Arte Precolombino: Yupanas and pallares: Stones to calculate
• Wikipedia: Yupana
• YouTube: Yupana Inka - Competencia Método Tawa Pukllay vs Método Arábigo