Very nice! Wythoff’s nim allows players to remove coins from the left pile, the right pile, or the same amount from both piles. This corresponds to moving down towards the x-axis, left towards the y-axis, or diagonally. As one of your references says, this is like moving a queen south, west, or southwest, where the first player to reach the origin wins. …But since Neil’s procedure takes the first available position in each antidiagonal, is seems that his queens mark the losing positions in a variant of Nim where we can, in addition, transfer coins from the left pile to the right pile, corresponding to also letting the queen move (say) northwest. Has this variant of Nim been studied? - Cris
On Aug 20, 2016, at 5:53 AM, J.P. Grossman <jpg@alum.mit.edu> wrote:
This is a restatement of Wythoff's variant of Nim, which has been well-studied. Here are a couple references:
http://www.wisdom.weizmann.ac.il/~fraenkel/Papers/WythoffWisdomJune62016.pdf http://www.msri.org/people/staff/levy/files/Book56/43nivasch.pdf
J.P.
On Sat, Aug 20, 2016 at 1:13 AM, Michael Kleber <michael.kleber@gmail.com> wrote:
Oh hey, I remember working this out on a chess board in my grandmother's house in 1992! Your queens are marking the winning positions in the nim variant where you have only up to two piles and you're also allowed to remove the same number from each pile (=move diagonally). Surely this is enough for NJAS to fill in all the details :-).
--Michael
On Fri, Aug 19, 2016 at 2:41 PM, Cris Moore <moore@santafe.edu> wrote:
What happens if the queens are also knightriders (i.e. they can also attack on lines of slope +-1/2 and +-2?)
- Cris
On Aug 19, 2016, at 2:02 PM, Neil Sloane <njasloane@gmail.com> wrote:
Dear Math Fun, Take an infinite chessboard, and starting at the top left corner, scan it by upwards anti-diagonals, placing a Queen whenever you reach a square that is not attacked by an existing Queen, The top left portion of the board is Qxxxxx... xxxQxx... xQxxxx... xxxxQx... xxQxxx... xxxxxx... xxxxxx... xxxxxQ... ...... The n-th term of sequence https://oeis.org/A065188 tells which row contains the Queen in column n. (It is easy to show that every column eventually contains a Queen.) If you click the "graph" button in A065188 you will see that the points appear to lie on two roughly straight lines, of slopes phi and 1/phi, phi being the golden ratio! The question is, why? How do these queens know about the golden ratio? Presumably this is related to budding sequences and sunflower seeds.
Notes. Let c(n) = A065188(n). A199134 lists n such that c(n)<n (the points on the line of slope 1/phi), and A275884 lists n such that c(n)>=n (the line of slope phi). A275885 gives lengths of runs of consecutive terms in A193134.
Initially
I thought these lengths were always 1 or 2, which might have been significant, but that is false. A275888 gives first differences of A275884. Initially I thought these terms were always 1, 2, or 3, but that is also false. I can't correct these errors until Monday, because the OEIS is in read-only mode for a server upgrade.
I can send a larger picture of the chess-board to anyone who is interested.
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