Martin Gardner's "Mathematical Games" column that popularized this game also included a description of a variant: Brussels Sprouts (<a href="http://en.wikipedia.org/wiki/Sprouts_(game)#Brussels_Sprouts" rel="nofollow">http://en.wikipedia.org/wiki/Sprouts_(game)#Brussels_Sprouts</a>), played with small crosses instead of dots.<p>While appearing similar, in Brussels Sprouts the result is always fixed: if there are an odd number of crosses initially, the first player wins; otherwise, the second does.
As a ten year old I played a game like this one but where you chose a fixed grid, and took turns filling in the edges of the component small squares. The winner enclosed the most squares, I think! As a chorister in a C11th cathedral there was a lot of time to be spent not listening to sermons, so we devised (or learned from someone's dad) these fun games. I think we usually used an 8x8 grid.<p>Edit: here we are: <a href="http://en.wikipedia.org/wiki/Dots_and_Boxes" rel="nofollow">http://en.wikipedia.org/wiki/Dots_and_Boxes</a>
People who like this might want to try Simon Tatham's puzzles. (the author of Putty)<p><pre><code> http://www.chiark.greenend.org.uk/~sgtatham/puzzles/
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Bridges and perhaps Untangle might be most similar.