Yupana
A yupana (from Quechua: yupay 'count')[1] is a counting board used to perform arithmetic operations, dating back to the time of the Incas.
The accountants and treasurers of the kingdom are found in every city, town, or indigenous village...In addition to providing this brief description, Poma de Ayala drew a picture of the yupana: a board of five rows and four columns with each cell holding a series of black and white circles.
[6] Found at Caraz between 1878 and 1879, this table yupana differs from that of Chordeleg as the material of construction is the stone and the central octagonal compartment is replaced with a rectangular one; towers also have three shelves instead of two.
[2]Discovered in Northern Ecuador by Max Uhle in 1922, this yupana is made of stone and its compartments are drawn onto the surface of the tablet.
This yupana is the one that is closest to the picture by Poma de Ayala in Nueva Coronica, while having a line fewer and being partially drawn.
In 1931, Henry Wassén studied the Poma de Ayala yupana, proposing for the first time a possible representation of the numbers on the board and the operations of addition and multiplication.
[2] In 1979, Carlos Radicati di Primeglio emphasized the difference of table yupana from that of Poma de Ayala, describing the state-of-the-art research and advanced theories so far.
He also proposed the algorithms for calculating the four basic arithmetic operations for the Poma de Ayala yupana, according to a new interpretation for which it was possible to have up to nine seeds in each box with a vertical progression of powers of ten.
[10] Glynn, as Radicati, adopted Wassen's idea of full and empty gaps, as well as a vertical progression of the powers of ten, but proposed an architecture that allowed yupana users to greatly simplify the arithmetic operations themselves.
The last column in Glynn's yupana is dedicated to the "memory", a place that can hold up to ten seeds before they are moved to the upper line.
[11][12] The following table shows the number 13457 as it would appear on Glynn's yupana: In 2001, the Italian engineer Nicolino de Pasquale proposed a positional solution in base 40 of the Poma de Ayala yupana, taking the representation theory of Fibonacci already proposed by Emilio Mendizabal and developing it for the four operations.
The Spanish chronicles written of the conquest of the Americas indicated that the Incas used a decimal system and since 2003 the base 10 theory has been proposed as the basis for calculating both with the abacus and the quipu[14] De Pasquale has recently proposed the use of yupana as astronomical calendar running in mixed base 36/40[15] and provided his own interpretation of the Quechua word huno, translating it as "0.1".
[16] This interpretation diverges from all chroniclers of the Indies, especially Domingo de Santo Tomas[1] who in 1560 translated huno into chunga guaranga (ten thousand).
[17][18] Relying exclusively on Poma de Ayala's design, Florio explained the arrangement of white and black circles and interpreted the use of the yupana as a board for computing multiplications, in which the multiplicand is represented in the right column, the multiplier in the two central columns, and the product in the left column, illustrated in the following table: The theory differs from all the previous in several aspects: first, the white and black circles would not be gaps that could be filled with a seed, but rather different colors of seeds, representing respectively tens and ones (this according to the chronicler Juan de Velasco).
The multiplier is represented as the sum of two factors, since the procedure for obtaining the product is based on the distributive property of multiplication over addition.
According to Florio, the multiplication table drawn by Poma de Ayala with provision of the seeds represented the calculation: 32 x 5, where the multiplier 5 is decomposed into 3 + 2.
Florio notes, however, that the mistake could have been on the part of Poma de Ayala in the original drawing, in designing a space as being occupied by a black circle instead of a white one.
In October 2010, Peruvian researcher Andrés Chirinos with the support of the Spanish Agency for International Development Cooperation (ACEID), revised older drawings and descriptions chronicled by Poma de Ayala, and finally deciphered the use of the yupana: a table with eleven holes which Chirinos calls a "Pre-Columbian Calculator", capable of adding, subtracting, dividing, and multiplying, making him hopeful of applying this information to the investigation of how quipus were used and functioned.
