It has long been a mystery of Aztec arithmetic: What is three arms plus five bones?
Now researchers know: five hearts.
The odd symbols had been noted for centuries -- thousands of them appear in Aztec property registries that were created around 1540. But no one knew the value of the symbols or how they were used to represent the size of land plots for tax assessment and other purposes.
After three decades of work, geographer Barbara Williams and mathematician Maria del Carmen Jorge y Jorge have found a solution that reveals a complex surveying system with a rudimentary ability to calculate the area of irregular shapes and manipulate fractional amounts.
"It cracks the code," said Williams, a professor emerita at the University of Wisconsin.
The researchers, who published their findings today in the journal Science, based their analysis on two books, called the Codex Vergara and the Codice de Santa Maria Asuncion. The manuscripts were written on paper brought by Spanish conquistadors, who had arrived in Mexico two decades earlier.
The researchers said the property drawings in the books were probably transcribed from even older documents written on tree bark or cotton cloth.
The pages of the books are filled with tiny property maps. For each plot, there are two drawings -- one showing the lengths of the sides and another showing the area. The measurements are represented by seven symbols: lines, dots, arrows, hearts, hands, arms and bones. Each map also includes the name of the property owner and the soil type.
Researchers already knew what each map represented and the value of some of the measurements. A line, for example, was the standard unit of length, which was known as a tlalquahuitl, or rod, and in modern units would measure a little more than 8 feet.
When the researchers knew the values of the units in roughly rectangular plots, they could easily follow the logic of the Aztecs and reproduce their calculations by multiplying lengths and widths.
But they were stymied in calculating many plots because they didn't know the value of the units. The breakthrough came when Jorge y Jorge, a professor at the National Autonomous University of Mexico, found that the values of some areas were prime numbers.
That meant that some of the unknown symbols had to represent fractions of a rod, she said.
By trial and error, she decoded the system.
A hand equaled 3/5 of a rod, an arrow was 1/2 , a heart was 2/5 , an arm was 1/3 , and a bone was 1/5 .
A set of at least five formulas emerged showing how the Aztec surveyors determined the areas of irregular shapes. In some cases, the Aztecs averaged opposite sides and then multiplied. In others, they bisected the fields into triangles.
Of the 369 plots the researchers examined, they could accurately reproduce the Aztec math in 287 cases, according to the study.
Still, Williams and Jorge y Jorge don't understand how the Aztec surveyors decided which formula to use for each area calculation.
In addition, it is unclear whether the same system was used in other city-states and if it applied to measurements besides land dimensions.