From: cet1@cus.cam.ac.uk (Chris Thompson)
Newsgroups: sci.math,rec.games.abstract
Subject: Re: 3 D four-in-a-row
Date: 25 May 1996 13:50:10 GMT

In article <vinkers-2405961133570001@bismuth.chem.vu.nl>, vinkers@chem.vu.nl 
(A Friendly Chemist) writes:
|> Dear forum,
|> 
|> I'm not sure if this is the right newsgroup 
|> to pose my question to, but I can't think of 
|> a better one except the nice but all too quiet 
|> alt.math.game_analysis I created on my own site ;-)

rec.games.abstract might be appropriate. I've added it to
the newsgroups.

|> I'm wondering about a game I play often lately: 
|> 3 D four-in-a-row.
|> 
|> It's just like normal four-in-a-row (i.e. to 
|> get four in a row is to win), only the playfield 
|> is composed of four squares above eachother each 
|> of which consists of 4x4 squares. I guess it's 
|> obvious in which ways one can make the four-in-a-row 
|> (difficult to explain in words ;-): within a 4x4 
|> square horizontal, vertical or diagonal, *or* between 
|> the four squares (from top to bottom) above eachother, 
|> straight diagonal or diagonal-diagonal (11, 22, 33, 44).

I take it you are indeed saying that all the usual lines
of 4 count: 48 parallel to the cube edges, 24 parallel to
the face diagonals, and the 4 body diagonals.
 
|> Could somebody tell me what would be a good strategy ?

The 2-player version is known to be a win for the first player. 
This was first proved by Oren Patashnik. His strategy involves 
thousands of opening lines, and I doubt whether there are [m]any 
human beings capable of executing it flawlessly without access 
to the database in an external form.

|> I suspect the corners in the top and bottom 4x4 
|> square to be important, for they create more 
|> possibilities than the other squares.  Creating 
|> more than one way to win with a certain move surely 
|> is great. 

There is a symmetry of the game you haven't noticed yet: the
inside-outside one. Label the positions with Cartesian coordinates 
(x,y,z) in the range 0 <= x,y,z <= 3. Then consider the map that 
changes any coordinate that is 0,1,2,3 to 1,0,3,2 respectively. 
("XOR each coordinate with 1", as a Computer Scientist would say.) 
You will find that this turns any line-of-4 into another.

Thus the 8 cube corner positions and the 8 central positions are
equivalent under this mapping, and are equally good as a starting
move. It would be interesting to know whether the game is still a
win for the first player if her first move is required to be on one
of the 48 remaining [equivalent] positions.

|>           On the other hand I'm playing with three 
|> (use three different coloured pencils for a beautiful 
|> view), so this isn't enough to be certain to win 
|> (when playing with two it is however).

I don't think I have ever seen the 3-player version in action.
You will have to consider the possibility of alliances: almost
certainly an alliance of two players will be able to ensure
that one of them wins and the third player won't be able to do
anything about it.

|> I invite you all to give it some thoughts. 
|> My Usenet access is pretty worthless nowadays and 
|> though I'm aware it's not good netiquette <tm> I'd 
|> appreciate if you could post replies also to my 
|> personal mail-adress : vinkers@chem.vu.nl

I don't think it is "contrary to netiquette" to request copies
of followups for such a reason [and I will e-mail you this one].
It's the "don't post, just e-mail me [the answer to my homework]"
that often raises hackles. "E-mail and I will summarise responses" 
is sometimes acceptable.

Chris Thompson
Email: cet1@cam.ac.uk