Friday, May 29, 2015

Carrollian Algebra

Alice talking to White and Red Queens
Alice's Adventures in Wonderland and its sequel Through the Looking Glass are some of the well-known children's classic tales written by Lewis Carroll, whose real name is Charles Lutwidge Dodgson, a mathematics lecturer at Oxford University. 

There are various algebra games designs embedded in Carroll's work. Let's use a few examples to illustrate. 

The first example is quiz-based design that comes from an excerpt in Through the Looking Glass, by Lewis Carroll:

'Can you do Addition?' the White Queen asked. 'What's one and one and one and one and one and one and one and one and one and one?'

'I don't know,' said Alice. 'I lost count.'

'She can't do Addition,' the Red Queen interrupted, 'Can you do Subtraction? Take nine from eight.'

'Nine from eight I can't, you know,' Alice replied very readily: 'but—'

'She can't do Subtraction,' said the White Queen. 'Can you do Division? Divide a loaf by a knife—what's the answer to that?'


Both Alice and the reader are entertainingly quizzed by the Queens. Try arguing with arithmetic!

The second example is puzzle-based design that comes from an excerpt in The Hunting of the Snark, by Lewis Carroll: 

Taking Three as the subject to reason about --
     A convenient number to state--
We add Seven, and Ten, and then multiply out 
    By One Thousand diminished by Eight.

The result we proceed to divide, as you see, 
    By Nine Hundred and Ninety and Two:
Then subtract Seventeen, and the answer must be 
    Exactly and perfectly true.


In a way, the Carroll's puzzle-based design is similar to the Jinx Puzzle introduced by Harold Jacob in his book Elementary Algebra (1979)

In Carroll's puzzle game, an initial number (can be any and need not be Three) first gets manipulated through a finite sequence of arithmetic moves (the above first stanza) to become another numerical result. A second finite sequence of arithmetic moves (the above second stanza) is then carefully chosen to go from this numerical result back to the initial number. Thus, the two finite sequences are essentially inverse of each other.

In Jacob's Jinx Puzzle game, an initial number (can be chosen from a few possibilities) goes through a carefully-crafted finite sequence of arithmetic moves to become a particular desired numerical result, giving the impression that All roads lead to Rome.

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