This article is a rough translation from Spanish. It may have been generated by a computer or by a translator without dual proficiency.
The term yupana refers to two distinct classes of objects:
- table-yupana (or archaeological yupana): a system of trays of different sizes and materials, which are carved into the top of the device into geometric boxes. Seeds or pebbles were placed inside, presumably for performing complex arithmetic calculations. The first of these tables was found in 1869 in the province of Azuay ( Ecuador) and prompted systematic studies of these objects. All archaeological examples are very different from each other. 
- yupana of Poma de Ayala: a picture on page 360 of El primer nueva corónica y buen gobierno, written by the chronicler of the Indies Felipe Guaman Poma de Ayala, represents a 5x4 chessboard.  The picture, although having some similarities with the majority of table-yupana, presents several differences from these. Notably, all the trays are same-sized rectangles, while table-yupanas also have trays of other polygonal shapes or of differing sizes.
Although very different from each other, most of the scholars who have dealt with table-yupana have extended reasoning and theories to the yupana of Poma de Ayala and vice versa, perhaps in an attempt to find a unifying thread or a common method. The Nueva Coronica was discovered only in 1916 in the library of Copenhagen and that part of the studies on it were based on previous studies and theories regarding table-yupanas. 
Several chroniclers of the Indies described, unfortunately approximately, the Incan abacus and its operation.
The first was Guaman Poma de Ayala, who around 1615 wrote:
... They count through tables, numbering from one hundred thousand to one hundred and from ten thousand and ten, until the unit. They keep records of everything that happens in this realm: holidays, Sundays, months and years. These 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 in which can be seen a series of black and white circles.
The Jesuit father José de Acosta wrote:
... they take the corn and put one here, three there, eight from another part; they move from a box and exchanged three other grains from one to another to finally get the result without error— 
Father Juan de Velasco wrote:
... these teachers were using something like a series of tables, made of wood, stone, or clay, with different separations, in which they put stones of different shapes, colors and angular shapes— 
The earliest known example of a table-yupana was found in 1869 in Chordeleg, Azuay Province, Ecuador. It is a rectangular table (33x27 cm) of wood consisting of 17 compartments, of which 14 are square, 2 are rectangular, and one is octagonal. On two edges of the table there are other square compartments (12x12 cm) raised and symmetrically arranged one another, to which two square platforms (7x7 cm), are overlapped. These structures are called towers. The table presents a symmetry of the compartments with respect to the diagonal of the rectangle. The four sides of the board are also engraved with figures of human heads and a crocodile.  As a result of this discovery, Charles Wiener began in 1877 a systematic study of these objects. Wiener came to the conclusion that the table-yupanas served to calculate the taxes that farmers paid to the Incan empire.
Found at Caraz in 1878 - 1879, this table-yupana is different from that of Chordeleg as the material of construction is the stone and the central compartment of octagonal shape is replaced with a rectangular one; towers also have three shelves instead of two. 
A series of table-yupanas much different from the first, was described by Erland Nordenskiöld in 1931. These yupana, made of stone, present a series of rectangular and square compartments. The tower is composed of two rectangular compartments. The compartments are arranged symmetrically with respect to the axis of the smaller side of the table. 
These yupana, made of stone, have 18 compartments of triangular shape, arranged around the table. On one side there is a rectangular tower with only one floor and three triangular compartments. In the central part there are four square compartments, coupled between them. 
Discovered in the province of Pisco ( Peru), these table-yupanas are two tables in clay and bone. The first is rectangular (47x32 cm), has 22 square (5x5 cm) and three rectangular (16x18 cm) compartments, and has no towers. The second is rectangular (32x23 cm) containing 22 square compartments, two L-shaped and three rectangular in the center. The compartments are arranged symmetrically with respect to the axis of the longer side. 
Discovered in the upper Ecuador by Max Uhle in 1922, this yupana is made of stone and its bins are drawn. It has the shape of a scale consisting of 10 overlapping rectangles: four on the first floor, three on the second, two in the third and one in the fourth. This yupana is the one that is closest to the picture by Poma de Ayala in Nueva Coronica, while having a line less and being half drawn. 
C. Florio presents a study  which does not identify a yupana in these archaeological findings, but an object whose name is unknown and which has been forgotten. Instead, this object is to connect to the tocapu (an ideogram already used by pre-Incas civilizations) called “llave inca” (i.e. Inca key) and to the yanantin-masintin philosophy. The scholar reaches this conclusion starting from the lack of objective evidences which recognize a yupana in this object, a belief that consolidated over years only for the repeat of this hypothesis never demonstrated, and by crossing data from the Miccinelli Documents and the tocapu(s) catalogued by Victoria de la Jara.
Fig. C - Really existing tocapu catalogued by Victoria de la Jara
Supposing to colour the different compartments of the table-yupana (fig. A), C. Florio identifies a drawing (fig. B) very similar to a really existing tocapu (fig. C) and catalogued by Victoria de la Jara. In addition, in the tocapu reported in figure D, also catalogued by V. de la Jara, Florio identifies a stylization of the tocapu C and the departure point for creating the tocapu “llave inca” (Inca key). She finds the relation between the table-yupana and the Inca key also in their connection with the concept of duality: the table-yupana structure is clearly dual and Blas Valera in “Exul Immeritus Blas Valera populo suo” (one of the two Miccinelli Documents) describes the tocapu we call Inca key as representing the concept of the “opposite forces” and the “number 2”, both strictly linked to the concept of duality.
According to C. Florio, the real yupana used by the Incas is that of Guáman Poma, but with more columns and rows. Guáman Poma would have represented just the part of the yupana useful for carrying out a specific calculation, which Florio identifies to be a multiplication (see below).
In 1931, Henry Wassen studied the yupana of Poma de Ayala, proposing for the first time a possible representation of the numbers on the board and the operations of addition and multiplication. He interpreted the white circles as gaps, carved into yupana in which to insert the seeds described by chroniclers: so the white circles correspond to empty gaps, while the blacks circles correspond to the same gaps filled with a black seed. 
The numbering system at the base of the abacus was positional notation in base 10 (in line with the writings of the chroniclers of the Indies).
The representation of the numbers, then followed a vertical progression such that the units were positioned in the first row from the bottom, in the second the tens, hundreds in the third, and so on.
Wassen proposed a progression of values of the seeds that depends on their position in the table: 1, 5, 15, 30, respectively, depending on who occupy a gap in the first, second, third and fourth columns (see the table below). Only a maximum of five seeds could be included in a box belonging to the first column, so that the maximum value of said box was 5, multiplied by the power of the corresponding line. These seeds could be replaced with one seed of the next column, useful during arithmetic operations. According to the theory of Wassen, therefore, the operations of sum and product were carried out horizontally.
This theory received a lot of criticism due to the high complexity of the calculations and was therefore considered inadequate and soon abandoned.
By way of example, the following table shows the number 13457.
Representation of 13457
This first interpretation of the yupana of Poma de Ayala was the starting point for the theories developed by subsequent authors, up to the present day. In particular, no one ever moved away from the positional numbering system until 2008.
Emilio Mendizabal was the first to propose in 1976 that the Inca were using, as well as the decimal representation, also a representation based on the progression 1,2,3,5. Mendizabal in the same publication pointed out that the series of numbers 1,2,3 and 5, in the drawing of Poma de Ayala, are part of the Fibonacci sequence, and stressed the importance of "magic" that had the number 5 for civilization the north of Peru, and the number 8 for the civilizations of the south of Peru. 
In 1979, Carlos Radicati di Primeglio emphasized the difference of table-yupana from that of Poma de Ayala, describing the state of the art of the research and theories advanced so far. He also proposed the algorithms for calculating the four basic arithmetic operations for yupana of Poma de Ayala, according to a new interpretation for which it was possible to have up to nine seeds in each box with vertical progression for powers of ten.  The choice of Radicati was to associate to each gap a value of 1.
In the following table is represented the number 13457
Representation of 13457
Glynn, as Radicati, adopted the same 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 to greatly simplify the arithmetic operations.
The horizontal progression of the values of the seeds in its representation is 1, 1, 1 for the first three columns, so that in each row is possible to deposit a maximum of ten seeds (5 + 3 + 2 seeds). Ten seeds of any row is correspond to a single seed of the upper line.
The last column is dedicated to the memory, which is a place where you can drop momentarily ten seeds, waiting to move them to the upper line. According to the author, this is very useful during arithmetic operations in order to reduce the possibility of error.
In the following table is represented the number 13457
In 2001, the Italian engineer Nicolino de Pasquale proposed a positional solution in base 40 of the yupana of Poma de Ayala, taking the representation theory of Fibonacci already proposed by Emilio Mendizabal and developing it for the four operations.
De Pasquale also adopts a vertical progression to represent numbers by powers of 40. The representation of the numbers is based on the fact that the sum of the values of the circles in each row forms a total of 39, if each circle takes the value 5 in the first column, 3 in the second column, 2 in the third and 1 in the fourth one; it is thus possible to represent 39 numbers, united to neutral element ( zero or no seeds in the table); this forms the basis of 40 symbols necessary for the numbering system. 
One of the possible representations of the number 13457 in the yupana by De Pasquale is shown in the following table:
The theory of De Pasquale opened, in the years after his birth, great controversy among researchers who divided mainly into two groups: one supporting the base 10 theory and another supporting the base 40 one. The Spanish chronicles of the time of the conquest of the Americas indicated that the Incas used a decimal system and that since 2003 the base 10 has been proposed as the basis for calculating both with the abacus and the quipu 
De Pasquale has recently proposed the use of yupana as astronomical calendar running in mixed base 36/40  and provided its own interpretation of the Quechua word huno, translating it as 0.1.  This interpretation diverges from all the chroniclers of the Indies, starting from Domingo de Santo Tomas  which in 1560 translated huno with chunga guaranga (ten thousand).
In 2008 Cinzia Florio proposed an alternative and revolutionary approach in respect to all the theories proposed so far. For the first time we deviate from the positional numbering system and we adopt the additive, or sign-value notation. 
Relying exclusively on the design of Poma de Ayala, the author explains the arrangement of white and black circles and interprets the use of the abacus as a board for making multiplications, in which the multiplicand is represented in the right column, the multiplier in the two central columns and the result ( product) is shown in the left column. See the following table.
The theory differs from all the previous in several aspects: first, the white and black circles would not be gaps that may be filled with a seed, but rather different colors of seeds, representing respectively tens and units (this according to the chronicler Juan de Velasco). 
Secondly, the multiplicand is entered in the first column respecting the sign-value notation: so, the seeds can be entered in any order and the number is given by the sum of the values of these seeds.
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.
The table multiplier drawn by Poma de Ayala with that provision of the seeds, represent according to the author, the calculation: 32 x 5, where the multiplier 5 is decomposed into 3 + 2. The sequence of numbers 1,2,3,5 would be casual, contingent to the calculation done and not related to the Fibonacci series.
Key: ◦ = 10; • = 1; The operation represented is: 32 x 5 = 32 x (2 + 3) = (32 x 2) + (32 x 3) = 64 + 96 = 160
The numbers represented in the columns are, from left to right: 32 (the multiplicand), 64 = 32 x 2 and 32 x 3 = 96 (which together constitute the multiplicand, multiplied by the two factors in which the multiplier has been broken down) and finally 151. In this issue (error) are based all possible criticisms of this interpretation, since 151 is obviously not the sum of 96 and 64. Florio, however, notes that a mistake of Poma de Ayala, in designing a black circle instead of a white one, would have been possible. In this case, changing just a black circle with a white one in the last column, we obtain the number 160, which is exactly the product sought as the sum of the quantities present in the central columns.
With a yupana as the one designed by Poma de Ayala can not be represented every multiplicands, but it is necessary to extend the yupana vertically (adding rows) to represent numbers whose sum of digits exceeds 5. The same thing goes for the multipliers: to represent all the numbers is necessary to extend the number of columns. Apart from the supposed error calculation (or representation by the designer), is the only one that identifies in the yupana of Poma de Ayala a mathematical and consistent message (multiplication) and not a series of random numbers as in other interpretations.
- Santo Tomas, "Lexicon o Vocabulario de la lengua general del Peru", 1560
- Radicati di Primeglio, "Il sistema contabile degli Inca: Yupana e Quipu", 1979
- Guaman Poma de Ayala, "Primer Nueva Coronica y Buen Gobierno", 1615
- José de Acosta - Historia Natural y Moral de las Indias - Libro VI cap XVIII (De los memoriales y cuentas que usaron los Indios del Perú)
- Juan Velasco - “Historia del Reino de Quito” - 1841 44, Tomo II, 7
- C. Florio, "Recovering memory - The Inca Key as Yanantin"
- William Burns Glynn, "Calculation table of the Incas", Bol. Lima No. 11, 1981, 1-15.
- Mora & Valero "La Yupana come strumento pedagogico alle elementari"
- Fiorentino, "La yupana elettronica: uno strumento per la didattica interculturale della matematica"
- N. De Pasquale "Il volo del condor", Pescara Informa, 2001
- Lorenzi, Incan counting system as easy as 1,2,3,5 (2004)
- N. De Pasquale, "The Saved Kingdom"
- N. De Pasquale, "Decimal Guaman Poma"
- C. Florio, "Incontri e disincontri nella individuazione di una relazione matematica nella yupana in Guaman Poma de Ayala", Salerno, 14-15 maggio e 10-12 Dicembre 2008 - Oédipus Editore, 2009
- Gilsdorf - Ethnomathematics of the Inkas
- Heliane Seline - Mathematics through cultures
- O'Connor & Robertson - Mathematics of the Incas
- (in Spanish) Poma de Ayala - El Primer Nueva Coronica y Buen Gobierno
- (in Spanish) José De Acosta - Historia Natural y Moral de las Indias
- (in Spanish) Velasco - Historia del reyno de Quito del America del Sur
- (in Spanish) Mora & Valero - La Yupana come strumento pedagogico alle elementari
- Leonard & Shakiban - The Incan Abacus
- (in Italian) Fiorentino - La yupana elettronica: uno strumento per la didattica interculturale della matematica
- (in Italian) Università Bocconi di Milano - La Matematica nelle civiltà pre-colombiane
- (in English) Incan counting system as easy as 1,2,3,5 - by Rossella Lorenzi
- (in Italian) Notizie sulla numerazione Inca e sulla yupana
- (in Italian) Un italiano scopre l'enigma della matematica inca
- (in Italian) Il Sole 24 Ore Domenica 10 Novembre 2002 – N. 308 – Pagina 35 - di Antonio Aimi - SCIENZA E FILOSOFIA Matematica precolombiana Scoperto il metodo di calcolo degli Inca
- (in Italian) L'unione Sarda - I numeri della natura nella scacchiera degli Inca - di Andrea Mameli
- (in English) "Guaman Poma Game, by N. De Pasquale, D. D'Ottavio