### Yupana - A Thesis by Ricardo Vilchez Chumacero (2013)

This thesis (written in spanish) contains detailed descriptions of the various hypothesis about the use of the Yupana. Among them, the hypothesis of Chirinos, Moscovich's hypothesis, and Pereyra's hypothesis. How to compute simple multiplications and divisions is also explained.

Asbtract
http://190.116.38.24:8090/xmlui/handle/123456789/55?show=full

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Pereyra's model is described from page 62 :

The value of a seed in a cell is the value of the number of circles of the cell multiplied by 1, 10, 100, etc. according to the position of the row of the cell.

I've used this model in most of my videos; this model of a Yupana is sometimes referred to as "Yupana dinamica".

The technique employed for additions and substractions is quite obvious.
The multiplication is said to use a very small multiplication table or no multiplication table at all.

The description exposes the distributive property of the multiplication on the sum (obviously) but doesn't show how practically one would compute a multiplication with one Yupana only.

Chirinos model is described from page 71 :

Globally, we find the description of a double model :
• in one of them, all the dots in a row have a value : from 1 to 11,
• in the other one, only a few dots in a row are used, with the values 1, 5, 10.
We will notice the use of the first row for the values  0.1, 0.2, 0.3, etc. instead of the row of the units (and obviously, this may be changed as is convenient).

Even for additions, one is said to find help in a table.

For multiplications, the requirement is the knowledge of the multiplication tables from 1 to 9 (which at first sight doesn't seem very interesting to me : we simply arrive at the solution that we use in the west : the only difference would be the complication of using seeds instead of signs, with no advantage nor help in calculus).

For divisions, you also need to use tables, and ... If you want to read the explanations, see pages 82 and 83.

The description of Moscovich's model begins in page 83 :

At first sight this description is unnecessarily complicated heavily using cultural concepts that seem without any practical use when we see how additions are described. (Or I didn't understand anything to the beauty of this model, which is possible).

The only useful (as far as counting is concerned) considerations that I see are :
• the seeds in the cells with one circle have the following values : 1 to 5 (five rows in column 1)
• those in the the cells with 2 circles have the values 10, 20, 30, 40, 50 (five rows in column 2)
• the seeds in the cells with 3 circles have the values from 100, 200, 300, 400, 500 (five rows of the 3rd column)
• the seeds in the cells with 5 circles have the values from 10000, 20000, 30000, 40000, 50000 (five rows of the column 4)
• no specific column for the thousands
• use seeds of different size for number that have no physical representation on the board [ I guess that means that 1000 can be represented in the 3rd column by one bigger seed, and 100000 represented by a bigger seed in the 4rth column ].
Computations are made, as usual, by moving seeds around according to their value, simplifying cells and columns that contain multiple seeds.

I think I'll try to use this kind of Yupana for my usual experiment on big multiplications. It may be handy.

Update 2014/11/16 : I tried ...  and at first ran into a problem because I was using only one type of seeds. In that case, as there is no column for the thousands, numbers that have a quite few thousands in them are ineffectively represented if all the seeds have the same value. For instance, 16000 is represented like 10000 + 12*500, that is (omitting the colors of the circles, that seem irrelevant to calculations in the examples given by Vilchez Chumacero) :

That's why bigger seeds have to be used. For instance with bigger seeds having ten times the values of normal seeds, 16000 would rather simply represented as :

And 16746 would be like that (not very pleasant because it requires thinking, but still usable) :
or for instance

The model of Pereyra is not simpler visually. For instance 16746 is (we keep the same orientation, to ease visual the comparison) :
or for instance

The first model of Chirinos may be visually the simplest. For instance, 16746 may be represented like that  :
or for instance