At the beginning of Chapter IX It's unclear which of the Queens is talking. Why did Carroll do it ? I think it is done as long as they talk to Alice together, having the same opinion, only once they start discuss between them ("and the Red Queen remarked") there is a point in differentiating between them.
Alice's Test Of Addition
The simple answer to the puzzle is 10, but I'm quite sure Carroll had more in his mind: I think that the hint is to the different counting basses. We usually count with the decimal basis but Mathematicians like most the Binary basis with only two numbers: Zero and One.
- 10 (Binary) is 2 (Decimal) so the answer "Ten" is a hint to "Two".
- 10 (Decimal) is 1010 (Binary)- Two times 10.
- Another answer to the puzzle is "Ten times One". Is there a way to convert between the basses ? the standard formula offers 210 - 1 or 1023.
- 1023 (Dec) is 1111111111 (Bin): Ten Times 1.
Alice's Test Of Subtraction
8 - 9 is, of course, above the Mathematical knowledge of 7 years old child. But, we can use the Looking Glass:
- By switching right and left, we come to 9 - 8 that Alice can do.
- If we turn the digits upside down, we come to 8 - 6.
- The difference between the two results above is 1 again.