The number of potential positions on a Go board
There's a reference to research done by Tromp and Taylor at Position that may be relevant. Their estimate is higher by many orders of magnitude. --Hyperpapeterie
Theorem: There are more than one googol (10^100) potential positions on a 19x19 Go board, allowing technically illegal positions in which a group of stones has no liberties.
Proof: Each point on the board either contains a black stone, a white stone, or no stone. There are 19*19=361 points on a Go board. Therefore the number of combinations is 3^361 > 3^360 = 3^(3*180) = (3^3)^180 = 27^180 > 10^180 > 10^100
Lemma: There are more than eight googol (8*10^100) legal board positions on a Go board. Why do I say eight googol instead of one googol? See my next theorem after this lemma.
Proof: Suppose you allow board positions with only black stones and no white stones. Each point either contains a black stone or no stone. The number of combinations is 2^361. However, exactly one of these board positions is illegal:
As you can see, this is illegal, because the huge black group has no liberties. But any other combination of black stones and empty points is legal. Therefore, there are 2^361 - 1 such combinations.
Let S be the set of legal positions allowing black and white stones. Let n be the number of elements in S. The set of legal positions allowing only black stones is a subset of S. Therefore, n > 2^361 - 1
-1 > -2^360
n > 2^361 - 1 > 2^361 - 2^360 = 2*2^360 - 2^360 = (2 - 1)*2^360 = 1*2^360 = 2^360 = 2^(10*36) = (2^10)^36 = 1024^36 > 1000^36 = (10^3)^36 = 10^(3*36) = 10^108 > 10^101 = 10*10^100 > 8*10^100
Theorem: There are more than one googol legal board positions that are unique after taking transformations into account. The following are examples of transformations:
Proof: Let T be a set of legal board positions that are unique after taking transformations into account. T is not the only such set, but it contains the maximum number of elements, which we'll call m. By replacing each element in T by eight elements that are the transformations of that element, the resulting set is identical to S. However, Some board positions are rotationally and/or reflectively symmetrical, meaning that they transform into themselves, Here are some examples: