# Freeing the trapped knight

The trapped knight problem is as follows. Place all natural numbers (starting with 1) in a spiral on an infinite square grid. A knight begins in the cell labeled 1. Each turn it jumps (in an L-shape) to an unvisited cell with the lowest number. It turns out that the knight eventually gets trapped - has nowhere to move. Neil Sloane made a great video about it.

Maarten Mortier simulated this game for all knights with movements $$(x,y)$$ such that $$1 \leq x,y \leq 80$$ and found that they all get trapped eventually. The knight $$(8,71)$$ takes the longest to get trapped at $$2400005$$ steps.

Now I have the following questions:

1. Is there a knight (x,y) that never gets trapped? Assume that $$1 \leq x \leq y$$.
2. Can we number the infinite grid in a different pattern, such that the standard (1,2) knight does not get trapped?
• Without restrictions on the pattern, 2 seems to be quite easy. Make a straight line for the knight to travel along with big numbers nearby so the knight follows the desired path. Commented Mar 10, 2021 at 6:06
• I was thinking of something like that, but I don't know how to formalize such a pattern of numbers. Any ideas? Commented Mar 10, 2021 at 6:08
• Here's a dense infinite tour for a standard (1,2) knight: math.stackexchange.com/a/2837147/207316 Commented Mar 10, 2021 at 7:39
• Nice that solves 2 very well. Commented Mar 10, 2021 at 9:15