Mathematics in the story
As Charles Dodgson was a mathematics professor, he incorporated some mathematical puzzles and jokes into the story.
When Alice is in the hallway, trying to figure out her identity, she tries to recite the multiplication table: 4 x 5 = 12 4 x 6 = 13 4 x 7 =… She notices it fails and comments that she’ll “never get to twenty at that rate”.
This could refer to the fact that multiplication tables traditionally stop with the twelves, so if you continue the progression until 4 x 12, you’ll only get to 19. (source: Martin Gardner, “The Annotated Alice”)
A more complex theory, is that Alice’s calculations are actually valid when you use a number system with a base of respectively 18, 21, 24, and so on (always incrementing the base with 3): 4 x 5 = 12 (base of 18) 4 x 6 = 13 (base of 21) 4 x 7 = 14 (base of 24) 4 x 8 = 15 (base of 27) 4 x 9 = 16 (base of 30) 4 x 10 = 17 (base of 33) 4 x 11 = 18 (base of 36) 4 x 12 = 19 (base of 39) However, then the system breaks down. In a number system with a base of 42, 4 x 13 is not 20. Therefore, Alice will indeed never get to twenty this way. (source: A.L. Taylor, “The White Knight”)