An examination of the "Trapped Knight" problem, as discussed in the Numberphile video "The Trapped Knight - Numberphile", found at https://www.youtube.com/watch?v=RGQe8waGJ4w.

Based on a comment by user "Smt Smt": "What happens if you mark 2084 as already visited before you start the game? Will it still get trapped somewhere else?" - https://www.youtube.com/watch?v=RGQe8waGJ4w&lc=UgxnOQujJFUbGTf13Tp4AaABAg

It appears that other users responding to that Numberphile comment have written code which does the same sort of thing as this code does. By looking at this discussion I was able to improve my code's efficiency considerably and also add some analysis of n-traps.

Interestingly, the sequence is non-monotonic but appears to be increasing on average. It would be interesting to look at the distribution of knight-stops probabilistically (what is the probability that the knight will stop at step n, or at value k, etc.)

There is a very interesting discussion in the YouTube comments section about "double traps", "triple traps", etc - I had trouble implementing the code recursively and this appears to be why (the idea of an n-trap is that, when your knight is trapped in a position, that position is an n-trap if you have to recurse n positions before you have another position you can go to).

Some questions that arise from the sequence:

• At any step k, is it guaranteed that we won't stop below some value f(k)?
• For some positive integer p, do p-traps become more or less common as we get further into the sequence?
• What is the probability of the knight becoming trapped at some step i, or some value x of the spiral?
• I found 1-traps, 2-traps, 3-traps, 4-traps, and 5-traps; are traps of any size possible? User Shirou97 found an 8-trap at step 754,165,534 - https://www.youtube.com/watch?v=RGQe8waGJ4w&lc=UgxnOQujJFUbGTf13Tp4AaABAg.8qVECW_mhGl8qWHlI1xgjd)
• As asked by Gareth Ma (https://www.youtube.com/watch?v=RGQe8waGJ4w&lc=UgxnOQujJFUbGTf13Tp4AaABAg.8qVECW_mhGl8qXLofMDB-R), what would happen if, rather than marking the positions of the stops as visited, we instead start the sequence from those points? User SBGif has analysed this in some detail (https://www.youtube.com/watch?v=RGQe8waGJ4w&lc=UgxnOQujJFUbGTf13Tp4AaABAg.8qVECW_mhGl8qZrgDGvqQQ and other comments in that chain).

Similar/interesting results were found in the comments section by the following users (with sincere apologies if I missed anyone - if I did miss you, please know that it was not intentional):

• Harrison Harris
• dLichti
• Darryl4779
• Shirou97
• John Merritt
• SBGif