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,

{b(j)} for your opponent).

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

advantage to either player (other than the advantages/disadvantages

which occur by chance if each player has a different starting

integer).

Thanks,

Leroy Quet

## Thursday, September 18, 2008

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