[eesmith]

The book "The Annotated Alice", mention by a couple of people here, says that is one valid interpretation, at https://archive.org/details/agt-annotated-alice-5807b6/page/... :

> The simplest explanation of why Alice will never get to 20 is this: the multiplication table traditionally stops with the twelves, so if you continue this nonsense progression—4 times 5 is 12, 4 times 6 is 13, 4 times 7 is 14, and so on—you end with 4 times 12 (the highest she can go) is 19—just one short of 20.

Gardner then writes "A. L. Taylor, in his book The White Knight, advances an interesting but more complicated theory" which is the changing base theory.

He ends with "For another interpretation of Alice's arithmetic, see "Multiplication in Changing Bases: A Note on Lewis Carroll," by Francine Abeles, in Historia Mathematica, Vol. 3 (1976), pages 183-84."

Available at https://www.academia.edu/download/122551204/82113901.pdf .