The Unconquered Mathematics of the Inca
By Professor Chris Staecker, PhD
Department of Mathematics
The quipu is a beautiful thing. It was a mystery, but we figured it out. It’s a system of knots tied in cords that the Inca and their predecessors used to keep track of numbers. The Inca had no written language, so the only records we have from the old days were written with fingers in quipus. Over 1,000 pre-columbian quipus still exist today, mostly in museums.
One old document, Felipe Guaman Poma de Ayala’s El primer nueva corónica y buen gobierno (1615), includes a famous drawing of an Inca recordkeeper—a quipucamayoc—holding a quipu. For historians, the quipus became a source of frustration: the old sources describe it, but none of them explain exactly how it worked. That’s where the mathematicians came in.
Left: Drawing of the Quipucamayoc, from El primer nueva corónica y buen gobierno by Felipe Guaman Poma de Ayala, 1615. He holds a quipu, with a diagram of the yupana at lower-left.
Right: A replica quipu on the wall of the author’s office in Bannow Science Center. (Come see it!)
In 1912, American mathematician and historian Leslie Leland Locke was the first to propose a specific interpretation. He believed, without much evidence, that the Inca were using a base-10 positional number system exactly like our own. So the number 74 would be represented by a 7-turn knot followed by a 4-turn knot. It seemed reasonable, if a bit familiar, but really there was no way to test Locke’s theory.
Above: The number 74, as it would be represented on a quipu. (Photo and knotting by the author.)
Later work by Marcia and Robert Ascher in the 1970s discovered that many quipu bundles include a special cord which shows the sum of all the others. So the quipu functions like a spreadsheet! Several numbers grouped together, with the sum written at the bottom. This work finally proved that Locke’s interpretation of the numbers was right: the correctness of the sums only makes sense if the Inca really were using Locke’s proposed system.
Anyone with an appreciation for mathematics will recognize a particular thrill here: we encounter mysterious symbols, see strange patterns, and apply mathematical analysis to figure it all out! This “decoding” of the quipu represented one of the first great successes in what we now call “ethnomathematics:” the study of the mathematical practices of traditional cultures.
But the quipu was not the only mathematical instrument used by the Inca. There was also the yupana, shown in the lower left of Guaman Poma’s quipucamayoc. It seems to have been some sort of counting board, where counters would be moved around to represent calculations—like an abacus without bars. However, details in the old sources are very scarce concerning the yupana, even more so than for the quipu. No one described exactly how it worked, and Guaman Poma’s simple illustration is the only drawing of a yupana that exists in any primary source.
But in the late 1800s, archaeologists began to find rectangular boards divided into compartments, often with different shapes and heights (see photos). These artifacts were quickly identified as the mysterious yupana, and again the mathematicians took an interest.
Above: Author’s photo of a “yupana” artifact in the Museum of Natural History, New York City. The museum describes this item as: “carved stone block of unknown purpose.”
Below: Author’s 3D-printed replica of the “yupana” artifact of Chordeleg, excavated 1869. (Come see it in my office!)
The yupana artifacts are beautiful objects, but they didn’t come with instructions. How exactly were they used as calculators? Over the decades, many people have announced their own “decoding” of these things. But serious scholarship has reached other conclusions.
The problem is, these artifacts don’t actually match the historical descriptions of the yupana. The striking feature of the artifacts is their three-dimensionality, but Guaman Poma’s drawing is a flat grid of squares. Additionally, the old texts describe the specific geographical region where the yupana was used, which doesn’t correspond to the locations where the artifacts have been found.
We also know that the Inca loved to use regular geometric shapes in their art, and they played board games. So these artifacts could very well be artistic carvings or gameboards. The scholars generally agree: the Inca probably did use a counting board, but they are lost today. And there’s no good evidence that these “yupana” artifacts have anything to do with calculation at all.
Unfortunately, the mystique of the Inca is too strong for this kind of dry academic conclusion. Especially in our time, a lifetime of serious study is no match for the idle notions of a confident hobbyist who “does their own research.”
A European engineer made headlines in 2004 when he announced his own “decoding” of the yupana artifacts. His explanation assumes a base-40 number system (the Inca used base-10). He boasted that it took him less than an hour to “solve the riddle,” and that he did so with no knowledge of the Inca culture (no surprise there). Media reports did not point out how deeply unserious it all was. This guy probably felt a certain cosmic kinship with the Inca, a respectful connection across the centuries when he finally cracked their code.
But it is not respectful to trample so casually on another culture’s ground. Respect begins with seeing the Inca not as a puzzle, but as a people. They were a complex people, and yes, a mathematical people, but they did not exist in order to be deciphered. They created their own mathematics for their own reasons, and their knowledge was not lost by accident: it was purposefully destroyed. And who are we, after all this time, to demand answers from the Inca? Their yupana, at least, will probably remain forever unconquered.
This article is adapted from the author’s YouTube video about the yupana: https://youtu.be/93QoXmIEsvw.
