Connect 4 is a fine, fun game. Defense is straightforward enough that tie games are frequent. Bigger homemade boards are fun and can open different strategies. 3d connect 4 is a pretty obvious derivation with the board being a cube.

One could play on a virtual board with more dimensions. Mathematically
one could do this, but the trick would be graphically simulating these dimensions to make the game more appealing to a general audience. How this simulation would actually take place is left as an exercise for the reader. The second trick would be stretching your head around how the 5th dimension interacts with the 4th and third so as to enable visualization of where pieces are in "space".

The game would track individuals and rank them according to wins thru various dimensions. Being a sixth dimensional master myself (with a decent track record in 7 and 8) I would not deign to play a master of a mere 4 dimensions. If this masterling would care to meet in 7 it might be a fine exercise. I will, however pass on 10 because only scary Russians and that AI from Caltech play there.

Graham's Number, explained by Ron Graham.https://www.youtube...watch?v=HX8bihEe3nA Looks like in this game of connect 4, the connections aren't in a line, but a series of connections that need to lie in a plane. [zen_tom, Oct 28 2014]

//How this simulation would actually take place is
left as an exercise for the reader.//

You would have to do it in the same way they depict
a hypercube - you choose a projection into three (or
two) dimensions. It's no different really from
showing an image of a cube in two dimensions, as
viewed from a chosen direction in three.

I've played 4x4x4 on a plastic grid (there are plenty of retail versions available, both 3x3x3 and 4x4x4; Google wouldn't give me a pic of the one I've used), and 3x3x3 and 3x3x3x3 on paper. You can easily "break" the extra dimensions and do as many as you like on paper; keeping track of it all is the hard part, but a virtual version would simplify that.
Going "virtual" would allow you to "switch" dimensions, to help with visualisation. Since we are accustomed to 3D (xyz), adding a 4th (w) gets difficult. You could line up 3 xyz lattices next to one another, but keeping an eye on alignments across the separated lattices is hard. So switch among the views: 1 lattice, but displaying whatever combo of 3 dimensions you want/need (maybe with the line-up of other lattices greyed in the background). Eg: xyz, wyz, wxz.
Going beyond 4D simply means you have more combinations of dimensions to look at. Eg: 5D = vwxyz, so look at vyz or wxy or vwy.
If I had access to MatLab, I could probably do a (very) simple experimental version, but alas... The numbers side of it is easy, it's the display and interpretation that's the hard part.

I may be off, but have a vague feeling that the definition
of
Graham's Number came about indirectly through an
investigation into the upper bounds of the number of
dimensions it takes before *any* arrangement of 4
particular
points in a grid to form a line. The lower bound has been
conjectured as being 6.

In other words, in a Graham-dimensional game of
connect-
4, whoever goes first should win, no matter what moves
they make.

[edit] See link, it's not exactly a connect-4 scenario - but
kind of related.

The trick here is that the more dimensions you add, the more degrees of freedom are added, and the more that you tip the balance towards player #1 if he defaults to going on the offense. To counteract, you'd have to change it to 'connect 5' or 'connect 6,' or somesuch.

I think it would be better to keep it to "tic-tac-toe" style with the winning line the full size of the board (be it 3 or 4 or whatever), rather than "connect 4" or "go" style, which is more open.
Ooh, just thought of an extra modification: toroidal boundary conditions ("go" having a much bigger playing area got me thinking) so there is no "edge" as such. Or would that make it TOO easy for #1 to win?

Probably it should be "connect n" with n some power of the number of dimensions.

One could have a temporal dimension with pieces that last a finite number of time periods in that place. The entire board with all spatial dimensions could be wheeled forward and backwards in time. A piece placed at time x would persist in that place thru x+1 and x+2, unless the space were already occupied at x+2.

Or the time x piece could have precedence and be able to unseat the x+2 piece - that makes earlier periods of the board more valuable because of the ability to change the future.

Is there only the one dimension that has gravity,
or in higher-dimension play do some of the new
dimensions get gravity as well? With 2D (standard)
Connect 4, the players have full control in one axis
which is at 0 deg and no control in the other at 90
deg. With more dimensions, the angle of gravity
between axes makes a difference, and once there
is an angle other than 90 degrees, the starting
point on these axes matter as well.

For example, in 2D, gravity is 90 degrees from one
axis, so it makes no difference where the player
puts the pieces on this axis. Pieces always fall to
the bottom. However if you create a simple 3D
version, then tilt it backwards at say 1 degree (and
modify the structure so that the pices can
somehow roll over each other at this low angle),
pieces will mostly fall straight to the bottom, then
roll to the back of the grid. If players can only
drop the piece in at the front row, then the
second layer can't start to fill in until the entire
bottom layer of that column is filled in. If however
they could choose to drop the piece into the
front, back, or any row in between, then a column
could fill in more randomly, but always increasing
in height toward the back. Alternately, if the
angle was increase to almost 45 degrees, but if the
players only was allowed to choose the starting
point on one of the tilted axes, there would be
relatively good control of piece placement initially,
but the far corner would not be accessible until a
ramp was formed.

[scad mientist], good point - a heirarchy of dimensions. It takes the design back to (as I mentioned above) "tic-tac-toe"-style versus "connect 4"-style.
A heirarchy would mean that you always place your piece at the maximum of the n-th dimension, somewhere (x) along either the (n-1)-th or 1st dimension (ie: your "placement" co-ords are (0,0,...,x,{max}) or (x,0,...,0,{max}) ). Then the heirarchy takes over to let your piece "fall" to it's final position.
This is getting both complicated and awesome, but it will need a better programmer than I to virtually build it. Any takers?

I have a simple Newtonian concept of gravitation. Maybe someone who understood gravitation from a relativistic perspective could better understand how it would act through additional spatial dimensions. Because that is hard for me.

I had this idea that dark matter could be explained by invoking a 4th dimension through which gravity propagates; dark matter is regular 3 dimensional stuff not in plane with us.

/surely/
Just so. And what would a 2D flatlander experience
as regards gravity from a 2D object outside his own
2D plane? In plane vector force from an invisible
intangible object.

/we would be seeing mysterious gravity from objects in other dimensions./

Just so! That is dark matter.
So which is more of a stretch - proposing additional spatial dimensions with the same rules as ours, or a different type of matter with properties different from normal matter?

But looking at Xavier's link, its the same old 2d tick tack
toe, supposedly enhanced, but not. Because if its only 3x3,
then the 2d winning/tie pattern is still there:

1. X starts. Puts X in middle of one of the 2d's. 0 does
whatever.

2. X puts second x in one of the corners -
a. If 0 had previously put the o in a corner of that 2d-tile,
then X puts it in the opposite corner.
b. otherwise X chooses any corner in that 2d-tile.

O's second move is anywhere on the board.

3.X response:
a. If O's last move was in a corner of the 2d-tile, X is
forced to put the x in between, O is then forced to
respond and put the o at the end of the new mid-line.
b. Otherwise, X wins by choosing any corner in the 2d-
space.

So in actuality we have 8 moves, that lead to a forced tie,
with no one winning in this "tile":
o1 o4 --
x3 x1 o3
o2 o4 x2

Now X has the advantage of choosing the next tile with an
x in the center.
O is forced to close off the third tile's center.

X can then put an x in the corner of the new tile
corresponding to an x in the first tile's corner.
Again O is forced to close off the third tile's center.

And X wins.

Regardless of the amount of extra dimensions available.

OK, that was a waste of time. I should have first read the m-
n-k link. Thanks xaviergisz!

Did I ever tell you that in
Amharic, one of the Ethiopian languages, Xavier means "The
almighty God"? In the morning you greet people with the
question: Indietta Derek? (How was your way) and the
answer: Gzavier Masgiv (The Lord is Great)