Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: FANO NIM
Replies: 1   Last Post: Apr 1, 2005 2:55 AM

 Messages: [ Previous | Next ]
 Glenn C. Rhoads Posts: 31 Registered: 12/13/04
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-:" | \
> / : _-' | `-_ : \
> / \' | `/ \
> / ,-' `. | .' `-. \
> /_-' `-..|..-' `-_\
> W--------------R--------------A
>
>
> 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.

Date Subject Author
3/30/05 Bill Taylor