It appears that the Olmecs–our final civilization in this series (1500-400 BC)–had a vigesimal, or base 20, counting system.
Counting is one of those things that you learn to do so young and so thoroughly that you hardly give it a second thought; after a few hiccups around the age of five, when it seems logical that 11=2, the place value system also becomes second nature. So it is a bit disconcerting to realize that numbers do not actually divide naturally into groups of ten, that’s just a random culturally determined thing that we happen to do. (Well, it isn’t totally random–ten was probably chosen because our ancestors were counting on their fingers.)
But plenty of societies throughout history have used other bases. The Yuki of California used base 8 (they counted the spaces between fingers;) the Chumash use(d) base 4; Gumatj uses base 5. There are also reports of bases 12, 15, 25, 32, and 6. (And many hunter-gatherer societies never really developed words for numbers over three or so, though they easily employed phrases like “three threes” to mean “nine.”)
The Yoruba, Olmec, Maya, Aztec, Tlingit, Inuit, Bhutanese, Atong, Santali, Didei, Ainu all use (or used) base 20. Wikipedia suggests that the Mayans may have used their fingers and toes to count; I suggest they used the knuckles+fingertips on one hand, or in a sort of impromptu place-value system, used the fingers of one hand to represent 1-5, and the fingers of the other hand to represent completed groups of five. (eg, 3 fingers on your left hand = 3; 3 fingers on your right hand = 15.)
Everything I have seen of reliable genetics and anthropology suggests that the Olmecs and Mayans were related–for example, one of the first known Mayan calendars/Mayan dates was carved into the Mojarra Stela by the “Epi-Olmec” people who succeeded the Olmecs and lived in the Olmec city of Tres Zapotes. Of course this does not mean that the Olmecs themselves developed the calendar or written numbers, (though they could have,) but it strongly implies that they had the same base-20 counting system.
You can compare for yourself the numbers found on the Tres Zapotes stela (above) and the Mayan numerals (left.)
In base-10, we have special words for multiples of 10, like ten, twenty, ninety, hundred, thousand, etc. In a base-20 system, you have special words for multiples of 20, like twenty, (k‘áal, in Mayan;) forty, (ka’ k’áal, or “two twenties;”) four hundred, (bak;) 8,000, (pic;) 160,000 (calab;) etc.
Wikipedia helpfully provides a base-20 multiplication table, just in case you ever need to multiply in base-20.
The Olmecs, like the Egyptians and Sumerians, produced art (particularly sculptures,) monumental architecture, (pyramids,) and probably had writing and math. They raised corn, chocolate, (unsweetened,) squash, beans, avocados, sweet potatoes, cotton, turkeys, and dogs. (It appears the dogs were also eaten, “Despite the wide range of hunting and fishing available, midden surveys in San Lorenzo have found that the domesticated dog was the single most plentiful source of animal protein,” possibly due to the relative lack of other domesticated animals, like cows.)
They also appear to have practiced ritual bloodletting (a kind of self-sacrifice in which the individual makes themselves bleed, in this case often by drawing sharp objects through their tongues, ears, or foreskins, or otherwise cutting or piercing these,) and played the Mesoamerican ballgame popular later with the Mayans and Aztecs. Whether these practices spread via cultural diffusion to other Meoamerican cultures or simply indicate some shared cultural ancestry, I don’t know.
Their sculptures are particularly interesting and display a sophisticated level of artistic skill, especially compared to, say, Norte Chico (though in its defense, Norte Chico did come earlier):
Frank Johnson, in his post Mystery Solved: Olmecs and Transoceanic Contact
A lot of people think the Olmec stone heads look a lot like Africans (and I can see why,) but–as lots of people have pointed out–they also look a lot like the local Indians who live in the area today, and so far I haven’t run across any genetic studies that indicate African DNA (which is quite distinctive) in any Native American population (aside from the DNA we all share from our common, pre-out-of-Africa ancestors, 70,000-100,000 years ago.) (There is one tiny isolated tribe over in Baja CA, [Mexico,] quite far from where the Olmecs lived, who do have some interesting DNA stuff going on that could indicate contact with Africa or somewhere else, but it could also just indicate random genetic mutation in an extremely isolated, small population. At any rate, they are irrelevant to the Olmecs.)
Frank Johnson, in his post Mystery Solved: Olmecs and Transoceanic Contact, goes through the laundry list of questionable claims about the Olmecs and does a great job of laying out various proofs against them. While I would not totally rule out the possibility of trans-Atlantic (or trans-Pacific) contact between various groups, just because human history is long and full of mysteries, the most sensible explanation of the origins and cultural development of Olmec society is the simplest: the Olmecs were a local indigenous people, probably closely related to most if not all of their neighbors, who happened to start building cities and pyramids.