## Thursday, September 18, 2008

### Nonconformity

2 players.
(Rules explained from the viewpoint of one player, you.)
Each player gets either (in one variation) their own
randomly-generated starting positive integer (m, for you, and n, for
your opponent), or (in the other variation) both players get the same
randomly-generated positive integer m.
m and n are >= 2.
Each player has a sequence of positive integers ({a(j)} for you,
a(1) = m.
b(1) = n (or m).
At each move, each player adds one new positive integer to their
sequence. This integer is picked secretly (ie. the k_th element of
each player's sequence is not influenced by what the other player
picks for his/her k_th element).
But after the integers are picked, these integers are revealed. (ie.
At each move, after the integers are picked, each sequence {as much of
each sequence which has been completed} is then known to both
players.)
So, the restrictions on the picked integers:
1) Each integer is >= 2, and
2) a(k+1) is a positive multiple of any divisor >= 2 of a(k).
(ie. a(k+1) is a positive multiple of any prime dividing a(k).)
(And likewise, b(k+1) is a positive multiple of any divisor >= 2 of
b(k).)
3) Each a(k) is not among {a(1), a(2), a(3),...,a(k-1)}.
(And likewise, b(k) is not among the previous terms of the
b-sequence.)
But the reason for the possible name (Nonconformity) is apparent
(somewhat) in this game's scoring:
You, for example, get a point as follows:
One point for each a(k) where:
min(b(j)) < a(k) < max(b(j)), for 1 <= j <= k-1.
and a(k) is not among {b(1), b(2), b(3),...,b(k-1)}.
(Likewise, your opponent gets a point for the counter-situation {where
all a's and b's are exchanged}.)
So, the scoring is based upon each a(k) and b(k),
and upon how these terms relate to the sequences up to a(k-1) and
b(k-1).
Players play a predetermined number of moves.
Sample game (with only 8 moves/player):
m=6, n=4:
player 1: 6, 4, 8, 10, 12, 14, 7, 21
player 2: 4, 2, 6, 3, 9, 12, 14, 7
In this (lame) example, player 1 gets one point for his/her 7, and
player 2 gets a point for his/her 9.
(A tie.)
(Of course, a game with more moves would be more interesting and less
likely to tie.)
I have played some a sample 'game' or two where I was my own opponent
(see above example). It seems as if there is a little stategy, at
least, to playing this.
What would be some good stategies for playing this game?
I have purposefully designed it so that the game gives no inherent