Suppose on an 8x8 board there is a white pawn on c5, and white' king is very far away, like h1. Where can the black king catch it?

The answer is found by drawing a square that includes c5 on the bottom-left from white's perspective. c5..c6..c7..c8.. now over the same number ...d8..e8..f8 then down to ...f7..f6..f5...and back over ..e5..d5..c5.

If it is the black king to move, he must be able to reach this square, or else white promotes and wins.

OK, now "the concept of the square" can be used in other ways. This geometric trick can be used to determine "any square", not just a promotion square, that must be reached in order to save a draw.

Because the gothic board is wider, there are usually fewer "crowning races" but more "horizontal opposition races" when it comes to K + P endings. Usually pawns are on each side of the board, somebody tosses one to get a king out of position, then races over to the other side to take them all out from behind. You can tell at a glance geometrically if you can resign or if you should run like hell after the king.

I think it is time for me to put up some more web page material. This is a topic that has no real equivalent in the 8x8 world.

--Ed