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Re: FANO NIM
Posted:
Apr 1, 2005 2:55 AM


Bill Taylor wrote: > Here's a cute little mathematical game. I'm surprised it hasn't been around > the abstract game world before. It's only the second game I know of that's > based on the Fano plane  the smallest possible 2D Projective Geometry. > > Here it is  it has vertices THEY WAR, and lines YEA WHY TRY HER RAW WET HAT, > neatly compiled into a triangle with three altitudes, and an incircle which > is also a "line"... > > Y > /\ > /  \ > /  \ > /  \ > /  \ > / _.""._ \ > /.'  `.\ > H.  .E > /: ".  .' :\ > /  ";T:"  \ > / : _'  `_ : \ > / \'  `/ \ > / ,' `.  .' `. \ > /_' `....' `_\ > WRA > > > Note there are no interior intersections between the altitudes and > the circle, only at the tangent points where they triply intersect > with the sides as well. > > _________________________________ > Anyway, the game is a form of Nim. > > The diagram starts with a small integer at each of the seven vertices, > the number of "seeds" at that vertex, preferably a different one for each. > (These can be decided on in one of several standard ways.) > > On each turn, the player to move must remove THE SAME number of seeds from > any three vertices in a line, (remembering that the circle is also a line). > > Last person to make a legal move wins. (This is the standard > winning condition for CGT games. There is also the "misere" version OC.) > """"""""""""""""""""""""""""""""" > > > I will leave it as a fun exercise for fans to compile a list, > hopefully exhaustive, of all the "N positions" (Next player wins), > and all the "P positions" (Previous player wins).
I generated the P positions (and by consequence the N positions) using a computer program and there are too many to list even for small numbers of seeds. First, any position in which the number of vertices with some seeds is <= 2 is a P position since no move is possible. Filtering out these, there are still 35 P positions where no vertex has more than one seed (if you consider the symmetries, then there are really only 6 distinct ones but the positions become harder to describe). Now if you filter out all positions where there is no move, we are left with the following (by "restricted" P positions, I mean those P positions from which there is at least one legal move, the number of seeds at any vertex is <= N, and at least one vertex has N seeds).
# restricted N P positions = ============ 1 0 2 315 3 2500 4 9430 5 27616
Obviously, memorizing a list of positions is not the way to go. The highly regular structure of the fano plane gives one reason to hope for a nice elegant way to compute the nimber of an arbitrary position.



