Packing Squares with Side 1/n
Packing Squares with Side 1/n
A finite volume of potatoes will fit in a finite sack. This seemingly simple statement leads to a family of very difficult questions, sometimes called potato sack problems.
Consider squares with sides , ,, …, . What is the smallest rectangle that can contain the squares as ? One bound is =-1, but no one has found a packing for a rectangle of that area. In 1968, Meir and Moser showed that a square of size × was enough. The current record is held by Marc Paulhus, who developed the packing algorithm used for this Demonstration.
1
2
1
3
1
4
1
n
n∞
∞
∑
n=2
1
2
n
2
π
6
5
6
5
6