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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?
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    $\begingroup$ 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. $\endgroup$ Commented Mar 10, 2021 at 6:06
  • $\begingroup$ I was thinking of something like that, but I don't know how to formalize such a pattern of numbers. Any ideas? $\endgroup$ Commented Mar 10, 2021 at 6:08
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    $\begingroup$ Here's a dense infinite tour for a standard (1,2) knight: math.stackexchange.com/a/2837147/207316 $\endgroup$
    – PM 2Ring
    Commented Mar 10, 2021 at 7:39
  • $\begingroup$ Nice that solves 2 very well. $\endgroup$ Commented Mar 10, 2021 at 9:15

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For 2, the spreadsheet below gives one approach. I am making a straight path for the knight to follow. I put 1,2,3 to start the path, then continued with odd numbers. I put the even numbers starting with 4 clockwise to fill the cells the knight could move to but I don't want it to.enter image description here

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