FANO NIM

[bill Taylor, 2005] 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).

And note, too, that if one too quickly learns the correct optimal play,
(though it is not as straightforward as regular Nim), one can easily
extend it to larger projective geometries; (the next one has 13 lines
comprising four vertices each and meeting four at a vertex.)

It can, OC, be played on any geometry at all; but the projective
nature of the game ensures that the winning condition is equivalent
to producing a line of zeros, which is a nice target to aim for!