What base is Alice dividing by in Alice in Wonderland?

Answer 1

After Alice grows in the corridor of doors, she starts to wonder if she still knows all the things she used to, so she decides to practise her multiplication table. She says: Let me see: four times five is twelve, and four times six is thirteen, and four times seven is--oh dear! I shall never get to twenty at that rate!

In his Annotated Alice, Martin Gardner points out that the simplest explanation as to why she won't reach twenty is that the multiplication table traditionally stops at twelve, so by the rule she's following, four times twelve will equal nineteen, whereupon she will be stuck.

However, A.L. Taylor, in his book The White Knight, suggests an alternative explanation. 4*5 is actually 12 in base 18. 4*6 is 13 in base 21. If this progression is continued, always increasing the base by three, the product will always increase by one until 4*13, when the system breaks down and a product of 20 cannot be reached.

4*5=12 (b18)
4*6=13 (b21)
4*7=14 (b24)
4*8=15 (b27)
4*9=16 (b30)
4*10=17 (b33)
4*11=18 (b36)
4*12=19 (b39)
4*13=1A (b42)

source: The Annotated Alice by Martin Gardner