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From: Mark Tilford on 27 Jun 2010 21:03
On 2010-06-21, Leroy Quet <qqquet(a)mindspring.com> wrote:
> 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.)
>
> Next, each player comes up with a rule for scoring points.
>
> 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.
>
> 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.
>
> Variation:
> 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 problems with this game?
>
> Thanks,
> Leroy Quet
>


A player could come up with a rule like


.... where (k = 1 and is to the right of, and on the same line as a 3, 6,
and 7, to the left of 2,2, 3, and 4, and below a 6) OR (k = 2 and is to the
right of on the same line as a 1 and 4, to the left of a 5,5,7,7,and 8) OR
....

And tailor each individual clause to specify any particular subset of
numbers as scoring. So ultimately, the game might as well be:

- Each player gives a score from 0 to n to every player (including
him/her self). Final score is the sum of all received scores.
From: Leroy Quet on 28 Jun 2010 10:58

Mark Tilford wrote:
>....
>
> A player could come up with a rule like
>
>
> ... where (k = 1 and is to the right of, and on the same line as a 3, 6,
> and 7, to the left of 2,2, 3, and 4, and below a 6) OR (k = 2 and is to the
> right of on the same line as a 1 and 4, to the left of a 5,5,7,7,and 8) OR
> ...
>
> And tailor each individual clause to specify any particular subset of
> numbers as scoring. So ultimately, the game might as well be:
>
> - Each player gives a score from 0 to n to every player (including
> him/her self). Final score is the sum of all received scores.


Maybe rules would be limited in how many letters/numbers they take to
write in English.
Or maybe the rules should be written down before the numbers are
placed on the line, but each player's rule is kept secret until after
all the numbers are placed.

Would either/both of these modifications work?

Thanks,
Leroy Quet

From: Leroy Quet on 30 Jun 2010 07:37
Leroy Quet wrote:
> 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.)
>
> Next, each player comes up with a rule for scoring points.
>
> 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.
>
> 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.
>
> Variation:
> 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 problems with this game?


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.

Can any of you think of a problem with the game even after the fix?

I didn't change this game so as to restrict the size of any rule.

Thanks,
Leroy Quet

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