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Subject: Counting unique openings in Tak rss

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Accounting for rotation and reflection, there are 12 unique positions after placing two stones on a 3x3 Tak board. Likewise, there are:
- 33 unique two-stone openings on a 4x4 board
- 85 on 5x5
- 165 on 6x6
- 315 on 7x7, and
- 518 on 8x8.

See the graphs for my calculations. The pattern this method of counting creates is rather nice.






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Very nicely done.
But how to proceed further?
And how to investigate about the usefulness of these openings in regard to winning the game?
 
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Bloody Gauntlet wrote:
But how to proceed further? And how to investigate about the usefulness of these openings in regard to winning the game?

I assume by doing something other than counting squares.

I just counted this because maths is fun. Also I'd seen other estimates thrown about that just seemed too large and I wanted clarity. I'm not sure how knowing this might help you play better.

One thing this limited analysis does show is not all first stone placements are equal, at least in terms of the possibility of memorising opening plays. There are essentially twice as many places to play the second stone if you play it off a centre line (diagonal or orthogonal) compared to playing on one, and four times more compared to playing on the centre space. That means some first stone opening plays are more susceptible to players playing memorised openings, as are the odd-sided boards if only because of the presence of a centre space and a centre row of orthogonal spaces.

Having said that, I'm not convinced this difference will make matter much given how large the possible play space becomes after playing even one more stone.
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