Tuesday, February 16, 2010

An Integer Sequence Game

This is a game for any number of players.

Needed: Pencil/pen, paper, calculator (with long display) perhaps. (Maybe this game could be played via a computer running the appropriate program.)

Start by writing down the integers 1, 2, 3,..., n, where n is at least 8 or more if the number of players is 2, I suggest. n is larger if there are more than 2 players.
This list of integers is called the "r-list".

The variable m starts the game with the value 1. In other words, m(0) = 1.

Players take turns. On the kth move (the kth move among all players together), the moving player lets r(k) = any uncircled integer from the r-list.
The player then circles that number.

m(k) is the value of m after the kth move.
Let m(k) =
r(k)*m(k-1) + (number of composites among m(0),m(1),m(2),...,m(k-1)).

Add to the moving player's score the largest value from m(0),m(1),m(2),...m(k-1) that divides m(k).

The move is complete when the moving player writes down m(k) at the end of the growing list of the values of m.

Players keep taking turns until k = n.


Example game, n = 8: (I may have made a mistake with my math.)
m(0) = 1
r(1) = 2; m(1) = 2*1+0 = 2. (Prime.)
Moving player gets 1 added to score.
r(2) = 8; m(2) = 8*2+0 = 16. (Composite.)
Moving player gets 2 added to score.
r(3) = 3; m(3) = 16*3+1 = 49. (Composite.)
Moving player gets 1 added to score.
r(4) = 5; m(4) = 49*5+2 = 247. (Composite.)
Moving player gets 1 added to score.
r(5) = 1; m(5) = 247*1+3 = 250. (Composite.)
Moving player gets 2 added to score.
r(6) = 4; m(6) = 250*4+4 = 1004. (Composite.)
Moving player gets 2 added to score.
r(7) = 6; m(7) = 1004*6+5 = 6029. (Prime)
Moving player gets 1 added to score.
r(8) = 7; m(8) = 6029*7+5 = 42208. (Composite, but this does not matter.)
Moving player gets 16 added to score.


How does the sequence {a(k)} begin, letting a(n) = the largest possible score for a 1-person game where the r-list contains the first n positive integers?

Leroy Quet

1 comment:

Veikko said...

Let n=8. The sequence for the maximum score is r(k) = {4,5,6,3,1,2,7,8},
from which m(k) = {4,21,128,387,391,787,5514,44118} yielding a(k) =
{1,1,4,1,1,1,1,387} summing up to 397.

The problem with this game seems to be that the winning strategy is impossible to visualize when the number of players is more than one.