16 Jan '15 19:46>1 edit
If I knew the rules for moving pieces in 3D chess, I'd try out the formula on that.
Can someone work up a Klein bottle chessboard? 🙂
Can someone work up a Klein bottle chessboard? 🙂
Originally posted by Paul Dirac IIMethinks I goofed on that one by double-counting the edges of the network.
For a rook on a standard board, E = 896. No matter what square you start a rook on, the degree d = 14.
M = 2(896)/14 = 128.
Same numbers hold if we put the rook on a cylindrical board.
Originally posted by Paul Dirac IIYikes, I goofed again, forgetting this time to apply the factor of 2 in the formula. Double those means for the queen on the standard board:
For a queen on a standard board, E = 728 (which is why I realized I had goofed on the number for a rook; a queen acts as rook + bishop) and d varies through 21, 23, 25, 27. This means M varies through
34 2/3
31 15/23
29 3/25
26 26/27
depending on the starting square.
Originally posted by Duncan ClarkePardon me blathering on about this problem, but I find it captivating. 🙂
Can this be solved as if it was a probability problem?
Originally posted by Paul Dirac IIUsing "S" as shorthand for the sum from n=1 to n=infinity, I see I can write the infinite series for the mean above as:
Can someone prove my series above converges to 4?
Originally posted by Paul Dirac III see my series is what they call arithmetic(o)-geometric:
This calculation is probably an exercise in high school analysis. I'll think about it some more.
Originally posted by Paul Dirac IIInstead of defining k as the number of squares on the board, let me define k to be the number of edge squares on a square board, i.e. a k x k board.
For a rook on a semi-infinite board, or a double semi-infinite board, or a fully infinite board, E = infinity and d = infinity.
M = 2(infinity)/infinity = indeterminate
I am going to conjecture that any time E and d are both infinite for some chess piece (we omit pawns which are irreversible, such that their network would be a directed graph rather ...[text shortened]... idered functions of the number k of squares on the board, and let k go to infinity.
Anybody?