# [seqfan] Stepping By Prime Increments In A Grid

Leroy Quet q1qq2qqq3qqqq at yahoo.com
Sun Jul 25 13:39:12 CEST 2010

```(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
[ ( [ ([( [ ( ([[o0Oo0Ooo0Oo(0)oO0ooO0oO0o]]) ) ] )]) ] ) ]

```

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