Imagine a multiplication table (like the one below, except it continues on forever):

1 2 3 4
+---+---+---+---+---
1| 1 | 2 | 3 | 4 |...
+---+---+---+---+---
2| 2 | 4 | 6 | 8 |...
+---+---+---+---+---
3| 3 | 6 | 9 | 12|...
+---+---+---+---+---
4| 4 | 8 | 12| 16|...
+---+---+---+---+---
|...|...|...|...|...

Find three of the same number in a straight line somewhere within the table. If this is not possible, show why not.

As I have noted,
you can draw a straight line
which passes through all three squares marked with a "4" in tomarken's
picture, although it doesn't pass through the center of the squares.

I suspect that these are the only three squares in the whole infinite
multiplication table where this is possible, but I don't want to do the
math.