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.)

Scoring:

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???

Thanks,

Leroy Quet

## Thursday, September 18, 2008

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