The "board" of this game is a regular n-gon drawn (as well as

possible) on paper, n being whatever the 2 players decide.

Each player uses a differently colored pencil and a straightedge to

take turns drawing 1 straight line per turn

between any 2 NONadjacent vertexes of the n-gon.

Lines must not coincide with any already-drawn lines (of either

color), but vertexes can be used more than once. (I suggest that there

be a limit of 2 or 3 lines per vertex, as to make things more

interesting.)

After a fixed number, m, of moves per player, scoring is as follows:

Player gets a point for each of her/his lines where the total number

of lines (of both players) crossing that particular line is an EVEN

integer.

(A shared vertex is not considered a line-interesection.)

The winner is the player with the most points, as you would guess.

If anyone decides to actually play this game, what values do you

suggest for m and n??

(Possible m's are, of course, bounded above by a function of n and the

maximum number of lines which may share one vertex.)

Thanks,

Leroy Quet

Update:

A few suggestions:

I think it might be best that, if n(k) = number of segments (drawn by

either player) crossing the k_th segment (drawn by the player in

question), then that player would get a point if n(k) is ODD, instead

of even.

I also think, for a more interesting game, that the number of sides,

m, of the polygon should be higher, at least 12. (m is limited by the

practical limitations of drawing all of this by hand, however.)

I also suggest that there be a maximum of 2 segments (total,

irrespective of which players drew the segments) that may be connected

to any particular vertex.

And I think that, in some ways, it would be best to have the number of

moves per player (before ending play) equal to floor((m-1)/2)

(ie. floor((m-1)/2)*2 total moves in a game), given a limit of 2

segments per vertex.

This leaves one or two vertex-pairs unconnected at the game's end,

which might have an effect on strategy (since the players cannot be

absolutely certain that there will be, at the game's end, 2 segments

connecting to each vertex that has not yet had 2 segments connected to

it at some point during play).

Please forgive my confusing lack of continuation with the uses of the

variable m (in 'm-gon' in the bottom) and the variable n (in 'n-gon'

in the first quoted post).

And then I use 'n(k)' to further confuse you all...

Thanks,

Leroy Quet

Update 2:

I think a interesting variation might be to, instead of each player

trying to get as many of their segments as possible having an even (or

odd) number of crossings, just have one player win if there are an

even number of *total* crossings, and the other player win if their

are an odd number of crossings.

(A 'crossing' is where 2 lines come together. For our purposes, if n

{n = 3 or more} lines come together at a single point, this counts as

n(n-1)/2 crossings.)

One advantage of this new variation is that we can play this game with

just one color of pencil/pen...

(which seems to actually be an issue when playing the game

in-practice...)

In this variation there are no scores, just a winner and loser, which

may be a disadvantage or an advantage, depending on your own opinion.

Thanks again, Leroy Quet

## Wednesday, September 17, 2008

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