Game Of Differences

Here is a simple game played, as many of my games,
with an n-by-n grid drawn on paper.
(I bet this is far from being the most fun game
I have made up. And there is a significant
chance I have submitted a similar game or the
very same game in the past.)
The game can be played solitaire or a 2-player
version can be played (see below).

Let us start with the 1-player version.
First the player puts a 1 in any of the grid's
n^2 squares.
On each move afterwards the player places the
integer k in any empty square next to (in the
direction of either above, below, left of, or
right of) the square with the (k-1) in it.

Play continues until either the string of integers
cannot be added to because there are no empty
squares next to the last integer written in the
grid, or until the final integer (n^2) is placed
in a square.

Now, there are 2*(n-1)*n different ways of pairing
off immediately adjacent squares (adjacent in the
directions of either above, below, left of, or right of).
The player's score is the sum of the absolute values
of the differences between each pair of immediately
adsjacent integers.

So, for example,
if we have the 4-by-4 grid filled in like this:

1 2 15 16
4 3 14 13
5 8 9 12
6 7 10 11

we have the score:

(Adding up the horizontal differences:)
1 + 13 + 1
+ 1 + 11 + 1
+ 3 + 1 + 3
+ 1 + 3 + 1
+

(Adding up the vertical differences:)
3 + 1 + 1
+ 1 + 5 + 1
+ 1 + 5 + 1
+ 3 + 1 + 1
=
64.

If at least one square of a pair is empty at the
game's end, the pair of squares contributes zero
to the score.

Variation:
We take the *product* of the differences between
adjacents, instead of the sum of differences, to
get the score.
(If at least one square of a pair is empty at the
game's end, the pair of squares contributes a
multiplication by 1 to the score.)

Question: Would the path of integers which gives
the maximum total sum also necessarily be the same
path which gives the maximum total product?

Another question: What is the integer sequence
where the nth term is the maximum possible score
for a n-by-n grid?
(I am really talking about 2 sequences:
where the score is derived by summing, and
where the score is derived by multiplication.)

2(or more)-player variations:
Variation 1) Just play the solitaire version of
same sized grids (and same rules) for each player,
and then the winner is simply the player with the
highest score.

Variation 2) One grid. Player 1's score is the sum
of differences for horizontal pairs;
Player 2's score is the sum of differences for
vertical pairs. Players take turns putting numbers
in the grid, so player 1 is odd, player 2 is even.

thanks,
Leroy Quet

.

Relevant Pages

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  • Game Of Differences
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