Diagonal, Non-diagonal, Almost Filling A Grid

Leroy Quet qq-quet at mindspring.com
Thu Dec 23 21:30:25 CET 2004

This question comes from my sci.math post at:

"Direction-Based Grid-Game (& Questions)":
(MathForum link because Google Groups links are even longer than before)

If we have an n-by-n grid, where n is an odd positive integer, we could 
fill in some of the squares of the grid as follows:

Fill in the center square.

Alternate filling in an *empty* square non-diagonally adjacent 
(immediately to the North, South, East, or West) to the last square 
filled, and diagonally-adjacent (immediately Northwest, Northeast, 
Southwest, or Southeast) to the last square filled.

So, start by filling a square neighboring the center square and above, to 
the left of,to the right of, or below the center square.
Next, fill in a square diagnonally adjacent to the last square filled.
Etc. Continue until no more squares can be filled in.

So, it seems to me to be impossible to fill in every square of a grid in 
this manner (other than the 1-by-1 case).
But what is the sequence where the n-th term is the maximum number of 
squares that can be filled in in a (2n-1)-by-(2n-1) grid?

Leroy Quet

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