Inefficiency in packing squares with unit squares

Translative Packing of Unit Squares into Squares

Packing Unit Squares in Squares: A Survey and New Results

Let s(n) be the side of the smallest square into which we can pack n unit squares. We improve the best known upper bounds for s(n) when n = 26, 37, 39, 50, 54, 69, 70, 85, 86, and 88. We present relatively simple proofs for the values of s(n) when n = 2, 3, 5, 8, 15, 24, and 35, and more complicated proofs for n=7 and 14. We also prove many other lower bounds for various s(n). We also give the ...

Packing Unit Squares in a Rectangle

For a positive integer N , let s(N) be the side length of the minimum square into which N unit squares can be packed. This paper shows that, for given real numbers a, b ≥ 2, no more than ab− (a + 1− dae)− (b + 1− dbe) unit squares can be packed in any a′ × b′ rectangle R with a′ < a and b′ < b. From this, we can deduce that, for any integer N ≥ 4, s(N) ≥ min{d√Ne, √ N − 2b√Nc + 1 + 1}. In parti...

On Packing Squares with Equal Squares

The following problem arises in connection with certain multidimensional stock cutting problems : How many nonoverlapping open unit squares may be packed into a large square of side a? Of course, if a is a positive integer, it is trivial to see that a2 unit squares can be successfully packed . However, if a is not an integer, the problem becomes much more complicated . Intuitively, one feels th...

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.