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Vi Hart's doodling videos and a 4 year old son interested in mazes has made me wonder:

What are some interesting mathematical "doodling" diversions/games that satisfy the following criteria:

1) They are "solitaire" games, i.e. require only one player; 2) They require only a pencil and blank sheet of paper; 3) They don't rely on abstract mathematical language/substitutions. More precisely, I'm interested in "picture" type games with very simple rules, and not the cheeky answer: "mathematics".

What I'd really like to see is a game with simple rules that, out of the planarity of a sheet of paper, somehow "generates" a maze for the player from the "automaton" type rules...A good starting point may be a solitaire version of dots-and-boxes or something like this...

If such a thing is impossible, under certain simple desirable assumptions, I'd like to see proofs of such a fact, too.

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This should probably be made Community Wiki. – robjohn Nov 17 '12 at 6:04
    
@robjohn: Already flagged it for CW....do you happen to know if it's possible to CW a question without moderator help? – Jon Bannon Nov 17 '12 at 18:55
    
If you don't see the Community Wiki checkbox when you start to edit the post, then you have to flag a moderator. I have converted your question to CW. – robjohn Nov 18 '12 at 0:14
    
I don't ever see the checkbox when I ask a question...is there anything I can do about this? – Jon Bannon Nov 18 '12 at 14:15
    
Evidently, the Community Wiki check-box on questions is mod-only, so you have to flag a moderator to make a question CW. – robjohn Nov 18 '12 at 16:45

You could draw an elementary cellular automaton with random initial row.

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Or run a turmite which moves on a grid of square cells, and never "decreases" the colour of a cell (where this colour is viewed as a number). Represent cell-colours as e.g. $\Box, \boxminus, \boxplus, \blacksquare$ (where every cell starts as $\Box$), so you never need to rub anything out.

For example ($r=90^\circ $ turn clockwise, etc.)

$\begin{array}{c|cc}\text{old} & \text{new} & \text{turn} \\ \Box & \boxminus & r \\ \boxminus & \boxplus & f \\ \boxplus & \blacksquare & u \\ \blacksquare & \blacksquare & f \end{array}$

If we call this $rfuf$ for short (from the above table's right-hand column) then the following are some more 4-colour turmites whose behaviour is not trivial: $rflf, rrlf, rurf, rulf, rlrf$. Unfortunately the most interesting turmites can both increase and decrease the number for a cell's colour, so running such a turmite on paper would need a pencil and a lot of rubbing out. For simplicity, I've kept to 1-state turmites so you don't have to keep track of the turmite's state as you doodle.

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