[Chinjut]

Re: "The multiplication that does not work", nothing in the quoted text seems to indicate that each multiplication should be interpreted in a different base, or anything like this. Certainly not that "four times [n]" should always have its result read in base 3n + 3 specifically.

It seems more likely to just be an absurd joke where Alice finds herself with an altered version of multiplication where 4n is interpreted as n + 7, causing multiplication to grow more slowly than normal, causing her to exclaim "I shall never get to twenty at that rate!" (a common exaggerated but non-literal use of "never", similar to "This is taking forever!" meaning "This is taking a long time!", not "This will literally never end").

The idea that we're instead supposed to think Alice thinks "four times 13 (decimal)" is to have its output read in base 42 (decimal), thus as "1A", considered distinct from "20", the latter being what would be "twenty", and thus she will literally never get to "twenty"... This just doesn't seem well-supported by anything in the text.

[j16sdiz]

The leading theory is they are quaternions.

https://www.tandfonline.com/doi/full/10.1080/26375451.2022.2...

https://www.newscientist.com/article/mg20427391-600-alices-a...

[Chinjut]

These links' mentions of quaternions are about a different part of the book (the tea party). Furthermore, even regarding that different part, your first link explicitly debunks the second link and disavows the quaternion connection the latter alleges. Your first link's whole point is to conclude "it is indeed very unlikely that Dodgson had the quaternions in mind when writing the tea-party chapter."

[Chinjut]

Why would you need to presuppose some inexplicably shifting number base to get the result of "four times [n]" always equaling n + 7? What does that get you over just more simply observing "For Alice, four times [n] has come to be n + 7"? Shifting number bases are a pointless supposition here. They don't explain anything better than what is already happening without them.

[raldi]

I agree. She’ll never get to 20 because kids only do the times table up to 12.

[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 .

[Xirdus]

Up to 12? Is that a British/Anglosphere/Victorian thing? In Poland they teach up to 10, which is suffinient for arbitrarily large numbers because they also teach long division and how to combine it with times table. Technically up to 9 would be sufficient but 10 is such a nice round number.

[raldi]

Yeah, every US sixth-grader can instantly tell you 12x12=144 but will have to puzzle out 2x13 the long way.

[gerdesj]

Gross!

[AnotherGoodName]

It is yes. The anglosphere has historically been somewhat base 12 in currency, time and units of measurement.

Currency is now metric but there’s still a few base 12 things in common usage (feet and inches) in the us at least. Nobody’s gone to metric time yet and base 12 transfers smoothly to base 60 too.

[Xirdus]

Of course it's because of imperial units. TIL, thanks. But on a sidenote, I question the utility of knowing x11 and x12 when working with time. x15 could be useful, unfortunate they don't teach that (but I think most people with higher education learn it on their own).

[acheron]

Feet and inches long predate imperial units, and the US has never used the imperial system, btw. “Imperial” has a specific meaning and isn’t just “anything not metric”.

Anyway, base 12 is also built into most Germanic languages which have unique names for 11 and 12 (rather than something along the lines of “one-teen” and “two-teen”, which is more common in Romance languages IIRC.

[zabzonk]

It's definitely a UK thing. And 12 is a nicer round number than 10 - ask the Babylonians!

[sltkr]

What's so nice about twelve-and-two (12)? Twelve (10) is a much nicer round number.

Though programmers may prefer base two (10) or base twelve-and-four (10).

[Xirdus]

If you say it in German, it doesn't even sound out of place. Zweiundzwölf, vierundzwölf.

[yread]

You're absolutely correct, the base is not specified. That's the joke. 1-1=0 would not be a joke. Perhaps it's better not to think of it as a joke. When mathematician reads what seems like nonsense, questions like "hmm is there a base where this would be true?" and "which bases is this true in?" pop up

[Chinjut]

I'm a mathematician, and my reaction here is somewhat different.

[yread]

So what's a good math joke?

I like this one: Young mathematician goes to first grade and the teacher asks who knows what is 1+2. She stands up and says "I don't know what is it, but I do know that it's the same as 2+1 as addition is commutative in the monoid of natural numbers"

[wzdd]

People said the same thing about a joke Douglas Adams made in his Hitch Hikers series -- that the (corrupted) Ultimate Question to which the answer was 42 ("what do you get if you multiply six by nine?") was a maths joke because 6x9=42 in base 13. Douglas Adams said this was nonsense.

[Xirdus]

Reminds me of one of my favorite joke programs.

  #include <stdio.h>

  #define SIX 1+5
  #define NINE 8+1
  
  int main() {
      printf("%d\n", SIX * NINE);
  }

[srean]

Didn't Linus #define THREE as something else in Linux source code.

Was that the 2-3-4 tree? Can't seem to find it now.

[Xirdus]

It was from some filesystem code. https://news.ycombinator.com/item?id=7296497

  unsigned three = 1;
  unsigned five = 5;
  unsigned seven = 7;

These actually get changed through pointers to consecutive powers of 3, 5 and 7 respectively. `three` is initialized to the 0th power of 3, but because only a single 1 is needed by the algorithm, `five` and `seven` are initialized to the 1st powers instead.

[BoardsOfCanada]

And by accident, 42 happens to be the first base after her multiplication gives the answer 19 here (when 20 would be expected), although it would produce an answer of "tenteen", not twenty.