Tuesday, August 31, 2010

Add/Residue Game

A game for any plural number of players:

Every player "gets" the numbers, twice each, of 1 to n -- where n is some pre-determined positive integer -- to play with; the integers 1 to n placed in each player's "mod pile", and the numbers 1 to n placed in each player's "add pile".

(A pile may simply be a list of integers that the player crosses off as each integer is played. Or if n = 13, for instance, and there are two players, then there can be 4-total actual piles where the numbers are represented by playing cards of certain suits.)

When the game begins, we have the initial value of the variable m, m(0), equal to 0.

The players take turns moving.
On a move, the player moving can choose a number from either their mod pile or their add pile. Let this number be c(k), when on the kth move of the game. Each number can only be picked once from a given pile. If playing with playing cards, place the card representing c(k) in the "used pile". Only pick from cards in your mod or add piles, not in the used pile.

On the kth move of the game, the player, depending on whether they chose their number from the add pile or the mod pile, sets m(k) to equal:

m(k) = m(k-1) + c(k), if c is chosen from the add pile.

m(k) = m(k-1) (mod c(k)), if c is chose from the mod pile.
0 <= m(k) <= c(k)-1.

If, and only if, c(k) was chosen from the mod pile, then m(k) is added to the moving player's score.

Play until the numbers from both piles of all the players are picked.
(There will be 2*n*(number of players) moves.)

Largest score wins.

Leroy Quet

Thursday, August 12, 2010

Subsequence -- A Number Definition Game

This is a game for any plural number of players.

In the first part of the game the players take turns, each player picking any one nonnegative integer to be appended to the end of a growing list of integers. The first part of the game is over when m integers are picked, where m is a positive integer agreed upon before the game by the players, and where m is a multiple of the number of players.

Let the finite sequence of integers be {a(k)}, 1 <= k <= m.

The players, in the second part of the game, take turns each coming up with a definition that describes some subsequence of {a(k)}. If the last integer in the subsequence defined by the previous player is a(j), then the currently moving player tries to find a definition that defines
(a(j+1), a(j+2), a(j+3),..., a(j+n)),
for some positive integer n. (j=0 when the first player first moves.)

The subsequence must be the first n consecutive terms of a sequence of integers defined by "the definition". The definition must be of the form "b(k) =..." (= an explicit function of k and/or of previous terms of b). The definition may only contain:any of the ten numerical digits
/ (divide, allowing fractional quotient)
\ (divide by expression following the \, then take the integer part)
* (multiply)
^ (that which follows the ^ is an exponent)
k (the index of the term), and/or
b (as in b(k-1) [= a previous term of {b(k)}], for use in recursions).

Again, a(j+k) = b(k), for all k where 1 <= k <= n.

The currently moving player gets added to their score on a move:

r*n - (the number of characters that occur in their definition after the =),

where r is some positive integer constant decided ahead of time by the players, such as r = 10.

When all m integers have been described, then the game is over.

Largest score wins.

(It should be noted that r should be small enough such that a player creating, say, a polynomial P(k) which outputs a(j+k), for all k where 1<=k <= m-j, would end up only losing points. If the first player to move creates such a polynomial for all k, 1<= k <= m, then they should receive a negative score, and all other players tie for first place each with a score of 0.)

Any problems you can see with the game? (Of course there are.)

Leroy Quet