Thursday, September 18, 2008

Plus/Times In A Grid

For some even positive integer m,
we have a m-by-m grid.
In this 2 player game, each player has (m^2/2) counters,
each counter numbered with a distinct integer from 1 to (m^2/2).
The players take turns placing the counters into the grid's squares in
any order the players wish.
(We do not actually need the counters, for players can simply write
the numbers in the grid's squares. But the counters make it easy to
know which numbers each player has already used, for each integer is
to be used one per player.)
(Or we can simply play this on a computer.)
One player is rows, the other player is columns.
For, say, rows, every set of adjacent integers, where each immediately
adjacent (to left/right) pair is coprime, is multiplied, then these
groups of multiplied integers are all added up to get the
row-player's score.
For columns, we do the same, but we consider immediately adjacent
pairs which are adjacent above/below for multiplication if coprime.
As to help explain what I mean, here is an example (of a game played
against myself without using any strategy):
(Who plays which number is unimportant.)
8 5 8 3
6 1 2 6
7 3 1 5
2 4 7 4
Rows gets:
8*5*8*3 +
6*1*2 + 6
+ 7*3*1*5
+ 2 + 4*7*4
Columns gets:
8 + 6*7*2 +
5*1*3*4 +
8 + 2*1*7 +
3 + 6*5*4
I would guess that higher m than 4 would be more interesting.

We can use other criteria other than coprimality when determining
which integers to multiply.
(One advantage of coprimality is that if 2 positive integers are
lower, then
they are more likely to be coprime than if they had been higher, which
{I feel} improves this game's strategy.)
Any interesting variations on this game???

Leroy Quet

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