[seqfan] Stepping By Prime Increments In A Grid

[Leroy Quet]

(I posted this to sci.math and rec.puzzles, but have yet to receive a reply. See the question within.)


This is a puzzle or, if you prefer, a solitaire game. 
Start with an m-by-m grid, where m is an odd composite. (It is 
impossible to fill the whole grid if m is prime or even; and the 
puzzle is trivial if m =1.) 
Place an x in any square to start. 
Thereafter take moves. 
On the kth move (where the (k+1)th x is drawn) move directly up, down, 
left, or right -- treating the grid as if it has toroidal topology -- 
by {the kth prime}-number of squares from where you last drew an x. 
You must land on a square without an x. Put an x in the square you 
land on. 
Continue as far as you can. 
The question: For which m is it possible to fill in the entire grid 
with x's? 
I am guessing -- but this does not rise to the level of being a 
conjecture -- that all m where m is an odd composite are completely- 
fillable. 
I haven't tried this puzzle myself for even the most simple nontrivial 
case of m =9, but only because I am too lazy to calculate the first 80 
primes by hand. 
Maybe someone can solve for a couple m's using a computer, or maybe 
you are less lazy than I am and will do this by hand. 
Thanks, 
Leroy Quet 
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