Experimental prehistory & various digressions: January 2019

The Chirinos-Hayakawa's Yupana

(I just invented this name, we'll see later if there is another name already in use for that thing)

Reminder about the Yupana

The Yupana, is said to have been a prehispanic counting tool / abacus used in the Inca empire. It is said to have been used, as many counting tools in the world, by moving various types of seeds or calculi on a board. The Inca counting board may have been a 4 column grid - according to one 16th century source, Guaman Poma de Ayala - with each column associated with a number of dots (1, 2, 3, 5). There is no definitive explanation of how to use a Yupana : no living ancestral knowledge, nothing clear (yet) found in historical records, only several modern interpretations and educational experiments in south american schools.

Notes : counting boards were still used in Europe, around the time of the conquest of the Inca empire by the spanish. Although the focus nowadays is set on the Guaman Poma 4 columns Yupana, various other tools are also called Yupana. Those that have been found have a number of cells carved into wood, stone or clay. With so few historical traces, nothing rules out that different types of Yupana / grid / counting board / abacus may have been in use simultaneously, for different purposes, different types of calculations or games. Particularly, as we know that the Inca empire was vast and diverse in cultures, languages and peoples.

Finding an efficient and "prehistoric" way to use a 4 columns Yupana

Important points for a viable interpretation of the yupana: paleo-compatible, efficient, reliable in complex computations, limited knowledge requirements (no multiplication tables), reduced number of manipulations, lower attentional requirements, lower memory requirements, possibility to track back errors.

A novel interpretation for the 4 columns Yupana

(I've not yet thoroughly verified if this interpretation has already been described before)

We can create another functional interpretation of the Yupana by mixing two systems :

the Andres Chirinos Rivera interpretation of the Yupana : basically, each dot in a row has a different value, from 1 to 11;
and a notation - not linked to Yupanas - demonstrated by Hayakawa Yoshinori on his youtube channel (早川吉則) : a notation that allows to blindly calculate, as with an abacus, as would do a computer, by using the successive powers of 2, with a very limited set of basic moves and almost zero knowledge requirements ;

The merge between Chirino's Yupana and Hayakawa's notation gives us a Yupana where :

each dot in a row represents a power of 2 : 1, 2, 4, 8, 16, 32, 64, and even more if we need it for special cases, up to 1024;

(if we needed even more, for special calculations we could increase the number of columns and use a "super-yupana")

each row, represents a power of 10, as required for a base 10 numeration system (Inca);
the first 4 dots in each row (1, 2, 4, 8) are sufficient to note any number in base 10;

3 other dots (16, 32, 64) are commonly used during calculations, particularly during multiplications
the remaining dots (128, 256, 512, 1024) may be useful in special cases;

for instance when we want to use fewer rows to note numbers ;
an example, extreme : 512256 may be noted with only 2 dots, on 2 rows.

The first experimentations show that the system is simple, "paleo-compatible", requires zero knowledge of multiplication tables, and need relatively few manipulations. But a high level of attention is still required from the operator. And there is still no obvious way to keep track of the successive operations, in order to find and correct an error.

Illustration 1 - values of the dots in the first row of the "Chirinos-Hayakawa" Yupana.

Video of a "big" multiplication with the Chirinos-Hayakawa's Yupana

Simpler version : Mendizabal-Hayakawa

I've tried a much simpler version, that could be called the Mendizabal-Hayakawa Yupana. It would give the values 1/2/4/8 to the columns, instead of the values 1/2/3/5.

After some experimentation, this Yupana appears to be easy to use, but the 1/2/3/5 marks in the cells are confusing and facilitate errors. We can still try to find a good reason for keeping those 1/2/3/5 marks in the cells. For instance by focusing on the great interest of the groupings by 2, by 3 and by 5 that occur frequently during calculations and that allow easy simplifications (5x8=40; 5x4=20 ; 3x4 = 12 ; 3x8=24).

Illustration 2 :