## Friday, September 19, 2008

### Math Cobweb

First, draw a regular r-gon on a piece of paper, where r is a multiple
of the number of players playing this game.
Each player then makes up math-functions, the same number of functions
per player for a total of r functions, which transform the integers m
and n (n is a positive integer) into an integer, new m.
For example,
m <= m + n,
m <= floor(|m|/n),
m <= d(|m| + n^2), d is number-of-divisors function,
m <= ceiling(sqrt(m*n)),
m <= numerator of reduced |m|/n,
m <= Fibonacci(|m|) - n,
etc.
(The functions can be appropriate for any level of math-ability.)
The functions are written around the perimeter of the r-gon, one
function per side.
Now, before play, each player *secretly* picks an integer and writes
it down without showing it to anybody else.
(This rule is subject to change.)
m begins with the value of 1.
Player 1 starts at any vertex of the r-gon and draws a straight
line-segment, using a straight-edge if necessary, to any side.
m is modified by the function at that side and by n (which equals 1 on
first move; see below on how n is calculated).
Players alternate drawing line-segments, starting each segment where
last player finished their segment, drawing the segment in any
direction (with restrictions), and finishing the segment at the first
previously drawn segment or edge of the r-gon encountered.
Segments coming off of a segment hitting another player-drawn segment
must pass the earlier segment, continuing on the opposite side from
the most-previously drawn segment.
Segments coming off of a segment drawn to the edge of the r-gon must
'bounce' (in either direction) back into the r-gon's interior.
Do not draw any segments to any vertexes (of the r-gon or where
previously drawn segments intersect).
(See the game "Tangle" for a similar set of
segment-drawing rules, plus for some ascii diagrams.)
Now n is equal to the number of segments in the previous "stretch".
(A stretch is the the collection of segments between any two sides of
the r-gon which are immediately connected by a path of segments
{without connecting in the mean-time to any other sides of the
r-gon}.)
Every time a side of the r-gon is reached, m is modified by the
previous m and by n and by the function at that side.
Also:
A stretch may intersect any particular segment at most once.
Each side of the r-gon can be visited any number of times, but the
game is complete when the last-to-be-visited side is finally reached.
The winner is the player(s) whose secret integer is closest to the
final value of m.
Alternative scoring: Maybe players can make up a different situation
for scoring/winning, such as: (for 2 player game) player 1 wins if the
number of distinct primes dividing the final m is odd, player 2 wins
if this number of primes is even.
thanks,
Leroy Quet