Mayan Numeration

The pre-Columbian Mayans developed a fairly sophisticated system of
numeration, primarily for the purpose of making calenders and keeping 
track of time.  (A concern for quantifying the passage of time, and
minding the calender, seems to have been a characteristic of many
primitive peoples, and prompted much of the early record-keeping.)
An example of a Mayan representation of a number is shown below:

           

The Mayans wrote their numbers vertically, with each "digit" being
represented by either a set of dots and horizontal lines or else a
symbol that looks (to me) like an empty bowl, which denotes zero (an
impressive invention of the Mayans, considering how many millenia it
took the people of the other hemisphere to think of it).  For the 
non-zero digits, each horizontal dash represents 5, and each dot 
represents 1, and these are simply addrf together to give the value
of the digit.  Thus each non-zero digit consists of from 0 to 4 dots, 
and from 0 to 3 lines, and these arrangements, along with the "empty 
bowl", give representations for every number from 0 to 19.  

Then they used a "place" system (another impressive invention), with 
the lowest digit signifying 1's, and the higher places signifying 
more powers of the base which was nominally always 20.  However, the 
system had one anomaly in that the denomination increased by a factor
of 18 instead of 20 when rising from the second to the third digit.
The presumed explanation for this is simply that since the Mayans
were mainly interested is counting days, and their basic annual
calender had 360 days, it was most convenient for the denomination
of the 3rd least significant digit to be (20)(18) = 360 instead of
(20)(20) = 400.

Of course, one consequence of this anomaly is that the possible
representations of a given number are not necessarily unique.  For
example, suppose we tip the Mayan numbers over, so the digits are
horizontally arrayed, and we use our numerals to signify the digit.
Then the decimal number 360 could be represented in the Mayan system
as either  (1 0 0)  or as  (18 0).  A nice feature of our more
conventional fixed-base representations is that they give a strict
one-to-one correspondence between the natural numbers and all the
possible permutations of a fixed set of digits.