HotIce: Hi, once I saw a 25 Handicap as:
Now what is the correct answer for 25 Stones?
(Original (Over nine stones handicap) page by WillerZ)
For the sake of completeness, I wondered what a 41-stone handicap should be (41 stones is about right for a 30kyu playing a 9dan pro). I reckon what I came up with is about right, but the question I couldn't answer was: in what order should the stones be added from 10 through 40?
How many various symmetrical 41-handicap-stone-placements are there?
That depends if you ask only for central symmetry or, as in the two diagrams above, for symmetry when you flip the board 90° as well. In any case, notice there must be one stone on the tengen.
In the first case, we can restrict the choice to putting 20 stones in the left half of the board, the other stones being placed by symmetry. Hence, this is choosing 20 stones among 180 positions, for a grand total of 175142105857592248012292655 possibilities.
In the second case, we can restrict the choice to putting 10 stones in a quarter of the board, the other being forced by symmetry. Hence, this is choosing 10 stones among 90 positions, or 5720645481903 possibilities.
The same calculus could be easily extended if you want some diagonal symmetry too.