This is a simple game that has the potential to go horribly wrong...
This game is for any plural number of players. Let the number of players be m.
Each player has a different colored pen/pencil/crayon.
Make a number line with the positions immediately beneath it labeled in order with 1 through m*n, where n is some positive integer decided ahead of time by the players.
The players take turns. On a PLAYER'S k_th move, he/she writes (with the pen/pencil/crayon of his own color) the number k just above any one of the empty positions along the number line.
After every player has written n -- after a total of m*n moves, and the number line is filled up -- the next part of the game begins.
(When the first part of the game is complete, every integer k occurs exactly m times on the top of the number line.)
But before writing down the numbers, each player comes up with a rule for scoring points. The players all write down their rules, and only reveal them after the number line has been filled with numbers.
Each rule completes this sentence:
A point is scored for a player for every integer in the player's color where _______.
The rule must be based on the position number (below the line) of the integer (above the line) being tested , and/or on the neighboring integers written during play (above the line).
A rule must NOT be based on the colors of the integers or on any external variables.
The rules may use any mathematics the players personally choose.
All the rules apply to all the players' numbers fairly.
In other words, the players EACH come up with a rule, and all the rules are used to test all the players' numbers, and the points obtained (in respect to all the rules) by each player are summed.
An example of some rules:
A point is scored for a player for every integer k in the player's color where _______.
* k is next to exactly one integer of opposite parity.
* k = the number of divisors of its position-number.
* k divides the sum of its immediate neighbors.
* k is coprime to the sum of all numbers to its left.
(My examples use basic number theory, but you can involve other branches of mathematics.)
Largest score wins.
Play on a grid instead of number line.
Involve the number of the column and/or the number of the row of each number being tested, as well as neighboring numbers, possibly.
Any unforeseen (by me) problems with this game?
Update: (6/30/10) I changed this game so that the rules the players choose are written down BEFORE the numbers are placed along the number line. The rules are kept secret until after the number line is filled with numbers.