A line in Connect Four

Question

When I was young, I was playing with a friend. Of course, the topic soon turned to mathematics (as it always does), and we got into an argument over something. We went on for some time, then he got bored of rhetoric and grabbed a Connect Four board off the games shelf.

Taking a marker from the stationery cabinet, he drew a single straight line, presented it to me. I admitted that he was right, and we moved on.

All these years later, I only have a foggy memory of the conversation. Can you work out what the argument was over, and how he made his point?

I'm pretty sure I was in Year 2/3 or so then, so use that as a guide...

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^{This story is not by any means historically accurate}

Answer 1

Addition plus geometry divides a game board

Your friend’s graffiti motivation might have come from page 69 of Proofs Without Words, Volume 1, by Roger B. Nelsen, displaying an image exactly like a Connect Four game with a zigzag line instead of straight. Either way, a line can split the game’s cells (Thank you, wildBillMunson.) into two...

...equal triangles with 21 cells each, and 21 just happens to be the 6^th triangular number.

                  21   =   1 + 2 + 3 + 4 + 5 + 6
                     =   6 ( 6+1)  / 2

Sure enough, an n × (n+1) rectangle divides neatly into two n × n triangles, each with rows that add up to 1+ 2 + 3 + ... + n cells.

Thus...

     1+ 2 + 3 + ... + n   =   half of that n × (n+1) rectangle
              =   n ( n+1)  / 2

This is a truly classic proof without words.

Answer 2

I think the argument was over the question

How many line segments connecting four circles (or the centres of said circles) can be drawn on a connect four board?

You may have said

$69$ consisting of

$24$ horizontal lines
$21$ vertical lines
$12$ NW-SE diagonals
$12$ NE-SW diagonals

While your friend would have argued that there are indeed

$75$

And won the argument by drawing something akin to the following line

Answer 3

A different proof and a different friend, named Georg

This is not the correct solution but does show, for variety, how a diagonal line in Connect Four resembles Georg Cantor’s diagonal proof that real numbers are uncountable.   So if your friend had been Georg, here is how his playing Connect Four might have meant more to him than just winning.

The opposite-colored row cannot exist because some disc in it differs from each row along the diagonal that produced it.   Interestingly related to Connect Four only, and to cover rows not crossed by the diagonal, any matching row would contain 4-in-a-row of your color and thus you would have won before Georg could make his final move.

That real numbers are uncountable means, in short, that they cannot be listed in any order, even with an infinitely long list.   Cantor’s proof of this directly resembles the Connect Four claim, as paralleled in the picture above.

Answer 4

Maybe it had to do with

Pythagoras' Theorem. You can draw the hypotenuse of a 3-4-5 triangle on a connect 4 grid. Not that a single example proves the theorem, nor does drawing the line actually prove that its length is exactly 5, but it's the only mathematical thing I can think of where a grid can come in handy.