I believe any of a large class of algorithms that fulfill each of the following three subgoals infinitely often will produce a "flat maze". Subgoal 1: Use the smallest unused odd number. Subgoal 2: Use the smallest unused even number. Subgoal 3: Fill the lefmost unoccupied cell. Each subgoal is free to use any large unused numbers, so each subgoal is easy to accomplish. The number of subgoal interleavings with each occurring infinitely often is uncountable, so I expect the solution space is uncountable. Best, - Scott

Hello math-fun and SeqFan,

I don't know if this is old hat -- if yes please ignore.

Rules for a new arrangement of the natural integers :

0) At the beginning there was a (semi)-infinite row of empty labels; 1) Then came the Pencil who wrote "1" on the first (most- left) one; 2) "1" meant: "write now whatever integer you want (but not used so far) on the label to the right of me at a distance of "1" step; [and, more generally, an odd integers means: "go 2k+1 labels to the right and write smthg new on it -- the label must be blank, though". An even integers means: "go 2k labels to the left and write...".]

Question: is there an algorithm to rearrange all the na- tural integers with those two simple rules (leaving no empty labels behind) ?

I have started this possible sequence (a dot means "still empty label"; the next label to be named has an "x"):

1 3 5 7 2 11 15 4 17 13 9 6 21 23 25 27 10 . . 8 . 12 14 . 17 19 . . . . . . . 20 . . 22 . . 24 . x 18 . 32 . . . . .

This sequence looks like an "flat maze" -- thus the name. The above has been done by hand -- this explains why the rows are 10 integers long: labels are easier to reach be- cause a vertical step (downwards) adds 10 to the count.

Best, =C9.