We have S, the unit square, and T, the unit-side equilateral triangle. We are to find some other shape A, and consider copies (to which Euclidean congruences have been applied) A1 … Am lying within S and pairwise disjoint (aside from boundaries) and B1 … Bn lying within T and pairwise disjoint (aside from boundaries) -- but we do not require that the union of the A’s be all of S, nor that the union of the B’s be all of T. And the puzzle is then to choose A and those congruences so that the ratio (total area of S and T left uncovered) / (area of A) is as small as possible (ideally, of course, zero).
Is that right?
(If so, then the original problem statement’s use of the verb “tile” confused the hell out of me.)
So is the following what you mean?
We have S, the unit square, and T, the unit-side equilateral triangle. We are to find some other shape A, and consider copies (to which Euclidean congruences have been applied) A1 … Am lying within S and pairwise disjoint (aside from boundaries) and B1 … Bn lying within T and pairwise disjoint (aside from boundaries) -- but we do not require that the union of the A’s be all of S, nor that the union of the B’s be all of T. And the puzzle is then to choose A and those congruences so that the ratio (total area of S and T left uncovered) / (area of A) is as small as possible (ideally, of course, zero).
Is that right?
(If so, then the original problem statement’s use of the verb “tile” confused the hell out of me.)
Yes, this is the case, I couldn’t have put it better myself! ;)