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### Connect 4 - Number of Winning Arrangements

```
Date: 6/20/96 at 8:55:40
From: Holy Family School
Subject: Connect 4 - Number of Winning Arrangements

Can you help us find the formula for the number of winning lines on a
7 x 6 Connect 4 board? Any clues that can point us in the right
direction will be most useful.
```

```
Date: 6/20/96 at 21:41:50
From: Doctor Chaos
Subject: Re: Connect 4 - Number of Winning Arrangements

If I understand your question, you are looking for the number of ways
you can get "4 in a row", horizontally, vertically, and diagonally.  I
don't know if there is a "formula" for this but you can definitely
explore the patterns on a 6x7 grid.  Here's how to start:

Look at how many times "4 in a row" fits in ONE ROW OF 6.  I see 3
Since there are 7 of these rows, that gives you 21 possibilities.

Now move on to the number of times you can fit 4 "4 in a row" in a
column of 7.  I see 4.  With 6 of these columns, you get 24
possibilities.

Follow the SAME idea with the diagonals slanting to the RIGHT and the
LEFT, and you'll probably come up with the same total I did: 69

Try it!

You can also develop your own "formula", by setting up grids of
various sizes in a sequence to see if a real pattern emerges.  It can
be a fun investigation.  If you find a "formula" that works, let us
know!

-Doctor Chaos,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations

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