Given square tiles of dimension 4.27" and 1.87" plus or minus .01" what grout widths are possible to make low order least common multiples?
Grout width can vary by 10% without being seen as different.
The idea is that N tiles of tile X + G (N-1) = M tiles of tile Y + G (M-1)
So that in a running pattern at least in one dimension NX tiles can be replaced by MY tiles. (There is always 1 less groutline than tiles.)
The goal is to find solutions where for a given pair of tile sizes X and Y, and a small range of G that M and N are small numbers.
This is not strictly speaking a diophantine equation, except that it is close to a diophantine equation in nature in that the desired solution is limited to integers, even though all the constraints are rationals.
I've been playing around it with a spreadsheet.
In general I've found less bad solutions if the tiles have a larger common factor f. I'm bastardizing common factor here. D = i * f where D is the dimension of the tile, and i is an integer. f can be floating. E.g. A 4.25" tile and a 1.75" tile have a common factor of 0.25, since both tiles are integer multiples of this. I've also found less bad solutions when the grout width is either a divisor or a multiple of this common factor.
Is there a name for this class of problem?
What keywords do I need to look up to find better algorithms for finding solutions?
Warning: Not a mathematician. Be gentle. I'm guessing at tags. Feel free to adjust.