[DonRotolo]

OK, let's imagine we have a chain that has a pitch of 0.375 inch. That means each link is .375 apart, as are teeth on a sprocket...but there are even and odd links (or 'innie' and 'outie' links). So really, a full "link" is 0.750. Get that?

So let's say you want an Integer (whole, not fractional) number of links. If we have a sufficiently large sprocket, we can say that:

1. The number of links on 1/2 of the sprocket is an integer (one lnk per tooth, right?)
2. If we imagine a point at the very top of the sprocket, it should coincide with the center of the sprocket. So for any chain going from top to top of two sprockets, if the centers are an Integer number of links apart, there must be an Integer number of links between the tops. And the bottoms.
3. If we have an integer for the sprocket half and an integer between the tops and bottoms, that all adds up to an Integer. See it?

I went to the first calculator and used 2 sprockets of 40 teeth each and a center distance of 37.5 inches. Using the above, I would expect 2 * 20 links on the sprockets and 100 links top and bottom, a total of 240 links...and that is what the calculator says it is. So the calculator seems to be working for me....