I did an interesting exercise from learning about endgames in Silman’s book, and it talks about how to shortcut your way to calculate whether the enemy king can stop your pawn from queening.

It is an amazing technique and is somewhat indicative of the mathematical beauty of chess as well.

If you see the position below, the way to calculate this is to create a 90 degree triangle from your pawn to the diagonal edge of the board, so here one that goes from a4 to e8 and e4.

If the king is within the triangle then you won’t be able to queen, and if the king is outside the triangle then you will be able to queen.

What’s amazing here is that in the setup I show here, the king closest to the pawn at b3 won’t be able to catch it because it is outside the triangle but the other two will be able to catch the pawn because they are right on the edge of the triangle.