Generalized Prince Rupert's Problem
The original Prince Rupert's problem can be phrased as: What the length of the side of the largest square that will fit completely inside a unit cube?
The answer is at least one - the unit square will fit inside the unit cube. But that is not the answer. It turns out that the answer is sqrt(9/8) = 3/4 sqrt(2) (for those that "rationalize the denominator"). This length is approximately 1.06 or about 6% longer than the side of a unit square. The area, 1.125, is 12.5% larger than the area of the unit square.
The generalized Prince Rupert's problem is: What is the length of the side of the largest m-dimensional cube that will fit completely inside a n-dimensional unit cube. In the original Prince Rupert's problem, m = 2 and n = 3.
I have been working with Greg Huber on this problem for a considerable time. Greg got me interested in the Generalized Prince Rupert's Problem when he was a postdoc and I was a graduate student in Berkeley in the 1990's.
Greg had found a number of insights into the general problem and I did extensive numerical experiments to compile data for cases beyond the current provable results. Based on this data, both Greg and I have drawn a number of additional conclusions - some provable and some conjectures.
My numerical data will be available here once we sort out the paper we are writing and decide how to present the data. Until then, thank you for your interest and patience.