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Non-uniform probability distribution of chess960 to smooth the transition from regular chess?

This is an idea I had today for easing the transition from classical chess to chess960 (the alarming drawing rate at highest level in classical chess makes me think about it from times to times).

In chess960 all 960 starting positions are equally likely to be picked. Therefore the regular position, that players know so well, has only slightly more than 0.1% (1/960) chance to happen.

IMHO, this is not ideal.

A position should be all the more likely to be picked as it is closer to the regular position. The regular position itself could be chosen to be picked at say a 50% chance, for instance (this is an arbitrary pick, but 50% sounds reasonable to me).

For the other positions, do this : create a function that measures the distance (as say the number of permutations) from the regular position and use it to build a probability distribution that makes familiar positions more likely than weird ones.

First step (the distance function) is not trivial, but it's probably not very difficult either. Any math-inclined person would like to look into it?

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