ALMOST SQUARING THE SQUARE: OPTIMAL PACKINGS FOR NON-DECOMPOSABLE SQUARES

ABSTRACT We consider the problem of finding the minimum uncovered area (trim loss) when tiling non- overlapping distinct integer-sided squares in an N × N square container such that the squares are placed with their edges parallel to those of the container. We find such trim losses and associated optimal packings for all container sizes N from 1 to 101, through an independently developed adaptation of Ian Gambini’s enumerative algorithm. The results were published as a new sequence to The On-Line Encyclopedia of Integer Sequences®. These are the first known results for optimal packings in non-decomposable squares.