Written in each square of an infinite chessboard is the minimum number of moves needed for a knight to reach that square from a given square O. A square is called singular if 100 is written in it and 101 is written in all four squares sharing a side with it. How many singular squares are there?
I just sketched this out on graph paper, not sure I'll have a great way to describe it without images, but here goes. Starting from O and working outwards, once you've filled in all the squares that can be reached in at most 4 moves, each additional move (M) populates squares in an octagonal, checkerboard ring that is 4 rows wide and with sides of M filledin squares. The squares in the outermost row of this ring will subsequently be surrounded on all four sides by the next largest number.
So for a given number of moves M, 8M squares will contain this number and be surrounded by squares containing M+1. So when M = 100, there will be 800 squares that are "singular."

Posted by tomarken
on 20210401 07:49:57 