How the Maya Counted

Their count is from 5 to 5 until 20, and from 20 to 20 until 100, and from 100 to 100 until 400, and from 400 to 400 until 8,000. And they made much use of this count for the trade of cacao. They have other very long counts, and which they extend in infinitum, counting them 8,000 twenty times, which is 160,000, and returning to 20 they double these 160,000, and then continue in this way doubling by 20 until they make an uncountable number: they count on the ground or on a flat thing.

Diego de Landa wrote this in the 1560s, a few years after he had burned the Maya books he found at Maní. Three of those books survive anyway, in Dresden, Paris and Madrid, and they are full of numbers: columns of bars and dots in black and red, sums that run into the millions of days. This article goes through the system piece by piece. Each part has something to try, and beside it the page it comes from: Landa's Relación, the colonial dictionaries and books of Chilam Balam, the Dresden Codex itself, and the two scholars who first worked out its arithmetic, Ernst Förstemann and Charles Bowditch.

Twenty, four hundred, eight thousand

Spoken Yucatec counts in twenties. The great Motul dictionary, compiled by Franciscans in the sixteenth century, gives the words for each step.

the number twenty: hunkal. … twenty twenties: hunbak, 400. twenty times 400: hunpic: 8000.

Hun is one, so hunkal is one twenty, hunbak one four hundred, hunpic one eight thousand. These are the units Landa lists, and the same entry gives hunkatun for twenty years. Each unit is twenty of the one below it. Landa adds that the counts were kept on the ground or on a flat surface, which fits what we find in the codices: a number is a stack of places, each one worth twenty times the place beneath it.

Three marks

The written numbers use three signs. A dot is one, a bar is five, and a shell is zero. Any number from 1 to 19 is some dots over some bars: seven is two dots over a bar, nineteen is four dots over three bars. Twenty cannot be written in one place, so it moves up: one dot in the second place and a shell in the first.

Because a place can hold only nineteen, the position of a mark says what it is worth. This is the same idea as our own place value, built on twenty instead of ten. Try it below: type a number, or press the buttons beside each place.

The two settings show the one irregularity in the system, which comes in the next section. In plain counting the third place is worth 400. In the calendar it is worth 360.

Zero, and how the scribes multiplied

A place-value system needs a way to say that a place is empty, or the stack falls apart: 400 and 20 and 1 would all be a single dot. The Maya had a zero, written as a shell in the codices and as several other signs in stone. Bowditch, working through the Dresden Codex in 1910, showed what it was by doing the scribe's sums. On Dresden 24 a series rises in steps of 2,920 days, and the fifth term has a zero in it. He works the multiplication the way the scribe would have to.

To perform the multiplication we should say, “Five times zero units give no units; put down the zero; five times 2 twenties give 10 twenties; put down the 10; five times 8 three-hundred-and-sixties give 40; but this is the same as 2 of the fourth place, with none left of the third place; therefore put down zero in the third place and 2 in the fourth place, as shown in the codex.”

So 2,920 is written 8 · 2 · 0 (eight 360s, two 20s, no units), and 5 × 2,920 = 14,600 is 2 · 0 · 10 · 0. The scribe carries just as we do, except that 18 in the third place makes one in the fourth, not 20. Press 14,600 in the number builder above to see it written out. On the monuments the zero is sometimes a face. Bowditch identified one form by the same reasoning, from what the number had to be for the date to come out.

the form of the head with a hand across the face must mean zero, since the day given in A6 is Ahau, and any other kin number (among those which we have heretofore found in such connection) if counted from Ahau, either forward or backward, would bring some other day than Ahau.

Eighteen in the second place: the Long Count

When the Maya counted days, they broke the pattern once. Twenty days make a winal, but eighteen winals, not twenty, make a tun of 360 days, close to a year. Above that the count is in twenties again.

To express higher numbers the Mayas used the vigesimal system up to the katun and beyond, as will be explained later, except in the second term. Thus, 20 kins = 1 uinal. 18 uinals = 1 tun. 20 tuns = 1 katun.

Twenty tuns make a k'atun of 7,200 days, and twenty k'atuns a b'ak'tun of 144,000. The monuments give dates as a count of days in these units from a fixed starting day, so that 9.9.2.4.8 means nine b'ak'tuns, nine k'atuns, two tuns, four winals and eight days. That starting day is written 13.0.0.0.0 and named 4 Ajaw 8 Kumk'u. The second name is its place in two other counts, the 260-day count and the 365-day year, both of which run alongside the Long Count.

Choose a date below to see it in every count at once. To turn a Maya day number into a date in our calendar you need to know which of our days was day zero. The standard answer, the Goodman-Martinez-Thompson correlation, puts it on 11 August 3114 BCE. Radiocarbon dates from carved wooden lintels at Tikal agree with it (Kennett and others, Scientific Reports, 2013). A variant two days later is in some astronomy software, and the converter lets you switch.

Gears, and fifty-two years

The 260-day count pairs a number from 1 to 13 with one of twenty day names. Both advance every day: 1 Imix, 2 Ik', 3 Ak'b'al, and so on, until 13 Ben is followed by 1 Ix. Because 13 and 20 have no common factor, every pairing occurs once before the cycle repeats, after 260 days. The 365-day year runs beside it, eighteen months of twenty days and five more at the end. A given day in both counts comes back only when both have come round, after 18,980 days, or 52 years. Landa describes how the four day names that can begin a year divided the years among four gods, which is the subject of another article.

The counts behave like meshing gears with one tooth for each day: a gear of 13 teeth, one of 20 and one of 365, all turning a tooth a day. The colonial books drew the k'atuns on a wheel; gears are a modern picture of the same arithmetic. Each gear has a gold mark that starts on the reading line. Turn them and see how long it takes for all the marks to come back to it together.

The same arithmetic explains three rows of numbers in the Book of Chilam Balam of Kaua, a Yucatec manuscript copied in the eighteenth century and photographed by Teobert Maler in 1887. The page lists the numbers that come round when one counts by months and by k'atuns.

There was another number that they call Katun, which served almost as a key to find and establish the Katuns according to the order of their movement. … Katuns: 13. 11. 9. 7. 5. 3. 1. 12. 10. 8. 6. 4. 2.

A k'atun is 7,200 days, and 7,200 is 553 thirteens with 11 left over. So each k'atun moves the day number on by 11, which in a count of 13 is the same as going back 2. The k'atuns were named by the day on which they ended, always a day Ajaw, and their numbers fall by two each time. The other two rows on the page come out the same way from 20 days and 360 days.

Whole numbers instead of fractions

Venus takes about 583.92 days to return to the same place in the sky as seen from the Earth. The Maya scribes had no fractions. What they had instead was the habit of looking for a number of days in which several cycles all come out even. Five Venus cycles of 584 days are 2,920 days, and so are eight years of 365. Five pages of the Dresden Codex are built on this.

Now 2920 = 5 × 584 = 8 × 365. But 583.92 days is the length of the synodical revolution of the planet Venus, and 365.25 days is the length of the solar revolution of the earth.

The table runs the 2,920-day count thirteen times, so that it also comes out even with the 260-day count. Bowditch shows that the whole table returns to the day it started from.

The whole thirteen rows will then cover 13 × 2920 days, or 37,960 days, or 8 × 13 = 104 years of 365 days, or 65 synodical revolutions of Venus.

37,960 days is 146 runs of the 260-day count, 104 years and 65 Venus cycles. Choose cycles below to see where they meet. Whole days cannot match the sky exactly: by the end of the 37,960 days the 584-day Venus count has run about five days ahead of the planet. The table has places that look like corrections for this, and how they work is still argued over.

A multiplication table, and the moon

Dresden 24, the page with the zero Bowditch worked through, is in effect a multiplication table.

On Dresden 24 this number 2920 is used as a factor or difference, the page affording a kind of multiplication table up to 13 times 2920, or 5.5.8.0.

The moon is harder, because a lunar month is about 29.53 days and no small whole number of months comes out even. The eclipse table on Dresden 51 to 58 lists 405 lunar months. Förstemann read how the scribe made it add up: months of 29 and 30 days grouped into runs of six, with a few runs of five.

Now the Mayas had determined the lunar revolution so exactly that they perceived the incompatibility of the period of 11,960 days with a multiple of lunar revolutions. They found that 405 lunar revolutions amounted approximately to 11,958 days, which is, in fact, the largest number on the second half of page 58.

At the modern length of the month, 405 months are 11,959.9 days. Förstemann goes on to show the steps the table uses: 177 days for six months (three of 29 and three of 30) and 148 for five, mixed so that the average month comes out close to the true one. The span of the table, 11,960 days, is also exactly 46 runs of the 260-day count, so the table starts again on the same day of the ritual calendar.

Serpent numbers

The largest numbers in the codex are painted inside the coils of serpents on pages 61 to 69. They are counts of more than twelve million days, more than 34,000 years, and each one lands on a chosen day in both the 260-day count and the year. Förstemann, checking them, noticed how closely they were related.

Further, it is to be noted that the largest of the eight numbers, 12,489,781, is separated from the lowest, 12,388,121 (i.e., the black number from the red one of the first serpent), by only 101,660. … 101,660 = 5 × 18,980 + 26 × 260 or 391 × 260 or 7,820 × 13.

The difference between the two numbers is a whole number of 260-day counts, so both land on the same day of that count. Förstemann ends the passage by asking how the person who made the table worked the values out. The method is the one this article has followed from the start: place value in twenties, with eighteen in the third place, and cycles that meet after a whole number of days.

What has been found since

Bowditch and Förstemann had the three codices and a few hundred inscriptions. Most of what has changed since then has come out of the ground.

The oldest dated day. In 2022 David Stuart, Heather Hurst, Boris Beltrán and William Saturno published two fragments of painted plaster from San Bartolo in Guatemala that carry the day 7 Deer, written as a bar and two dots beside a deer's head. Charcoal from the same sealed layer dates them to roughly 300 to 200 BCE, which makes them the earliest securely dated record of the 260-day count (Stuart and others, Science Advances, 2022). The count was already written down, bars and dots and all, some fifteen centuries before the Dresden Codex.

Older still, in the buildings. A lidar survey of 415 ceremonial complexes on the southern Gulf Coast of Mexico, built between about 1100 BCE and 250 CE, found most of them aligned to sunrises on dates spaced by multiples of 13 and 20 days, the units of the 260-day count. Ivan Šprajc, Takeshi Inomata and Anthony Aveni read this as the count in use centuries before any surviving inscription (Science Advances, 2023). Aguada Fénix, the largest early Maya platform, is among them.

Tables on a wall. At Xultun in Guatemala, in a small room painted in the early ninth century, Saturno, Stuart, Aveni and Franco Rossi found columns of numbers written on the plaster: lunar tables, and possibly tables for Venus and Mars, some five hundred years older than the Dresden Codex. One array of 4,784 days, 162 lunar months, is laid out much like the Dresden eclipse table (Stuart, Maya Decipherment, 2012; the report is in Science 336, 2012). The astronomy of the codices had a long working life behind it.

The 819-day count. Many Classic inscriptions give a fourth count, of 819 days, which is 7 × 9 × 13. Its purpose was long unclear. John Linden and Victoria Bricker proposed in 2023 that it was meant to be run twenty times: 20 × 819 = 16,380 days, which is a whole number of 260-day counts and of 780-day Mars cycles, and close to whole numbers of the other visible planets (Ancient Mesoamerica, 2023). Choose the 819-day count and Mars in the cycles above, or the third set of gears, to see where they meet. It is the same method as the Venus pages.

How the eclipse table was used. John Justeson and Justin Lowry argued in 2025 that the Dresden table began as a lunar calendar, fitted to the 260-day count through the 46 runs it contains, and was then reset at its end in ways that kept its eclipse warnings usable for centuries (reported in Sky & Telescope; the paper is in Science Advances, 2025).

Stuart's book The Order of Days is the readable account of how these counts fit into Maya ideas about time, written in the run-up to 2012. Its main point about that year is the one the arithmetic above makes: 13.0.0.0.0 was the completion of a count, and the counts kept going. Inscriptions at Palenque name dates thousands of years beyond it.

The English of Landa, the Motul dictionary and the Kaua book is the library's translation. Bowditch and Förstemann are quoted from their printed English. The Paris Codex and Brinton's Maya Chronicles, on the colonial count of k'atuns, are also in the library. The date converter uses the proleptic Gregorian calendar for all dates.