I thought of, and actually fiddled with, a generalization of ceiling squares to include all positive integers, excluding perfect squares, both even and odd. Though not a genuine problem with this, there are some tradeoffs that make me say “Why bother?”
- For a given perfect square ceiling s2, there will now be 2s-2 rows in the ceiling structure which, given that one still only needs s-1 columns, results in a rectangle that is not square.
- Given that less symmetric or aesthetically pleasing trait, the ease of simply dividing by 2 until one gets an odd number seems more than easy enough.
So keep it pretty, folks! Unless of course having at least one zero in every column appeals to you. (Does the absence of them in every other column of a ceiling square bug you that much?)
In my defense for even fiddling with the idea, it has led to another idea I think I do want to explore: Look at all the values of ceiling rectangles, which are in fact nothing more than the values of m mod n for n from s-1 down to 1, for all positive integers m whose integer ceiling is the square root of s, and express the factorization that results from finding the largest divisor in that set, i.e., the leftmost zero value, as a quadratic polynomial in x that gives m when x=s.
I am not yet convinced such a side-trip will generate enough additional insight to be paper-worthy. I’ve been wrong before. Stay tuned.