Unit squares intersecting all secants of a square

Efficient Packing of Unit Squares in a Square

Let s(N) denote the edge length of the smallest square in which one can pack N unit squares. A duality method is introduced to prove that s(6) = s(7) = 3. Let nr be the smallest integer n such that s(n + 1) ≤ n + 1/r. We use an explicit construction to show that nr ≤ 27r/2+O(r), and also that n2 ≤ 43.

Packing 10 or 11 Unit Squares in a Square

Let s(n) be the side of the smallest square into which it is possible pack n unit squares. We show that s(10) = 3 + √ 1 2 ≈ 3.707 and that s(11) ≥ 2 + 2 √ 4 5 ≈ 3.789. We also show that an optimal packing of 11 unit squares with orientations limited to 0◦ or 45◦ has side 2+2 √ 8 9 ≈ 3.886. These results prove Martin Gardner’s conjecture that n = 11 is the first case in which an optimal result r...

Optimal Packings of 13 and 46 Unit Squares in a Square

Let s(n) be the side length of the smallest square into which n non-overlapping unit squares can be packed. We show that s(m2 − 3) = m for m = 4, 7, implying that the most efficient packings of 13 and 46 squares are the trivial ones. The study of packing unit squares into a square goes back to Erdős and Graham [2], who showed that large numbers of unit squares can be packed in a way that is sur...

Translative Packing of Unit Squares into Squares

Packing equal squares into a large square

Let s(x) denote the maximum number of non-overlapping unit squares which can be packed into a large square of side length x. Let W (x) = x − s(x) denote the “wasted” area, i.e., the area not covered by the unit squares. In this note we prove that W (x) = O ( x √ 2)/7 log x ) . This improves earlier results of Erdős-Graham and Montgomery in which the upper bounds of W (x) = O(x) and W (x) = O(x(...