problem of the day

[loveschak]

Molly Mathematician has two closed rectangular boxes, Box A and Box B, which she notices have the same volume and the same surface area. Box A's longest edge is longer than Box B's longest edge. Which box's shortest edge is longer? The four plausible answers I can think of are:

• Box A's shortest edge is longer than Box B's.
• Box B's shortest edge is

Comments

[tharasix]

For the 3D case, we have V = xyz and A = 2xy + 2xz + 2yz where V and A are constant. Since we have two linear equations of three variables, the solution to the problem is in one dimension, giving us infinitely many solutions, so item 4 is incorrect. I'm not going to take the time to prove it, but the geometry in my head says that Box B's shortest edge is longer than Box A's.

For the 4D case, we have two equations of four variables, making a two dimensional solution space. That says to me that either case is possible, but once again, I'm not going to try to prove it.

[loveschak]

You're definitely right about there being an infinite number of solutions. But, counterintuitively, the box with the longest edge also has the longest shortest-edge. For example, the box with dimensions 1,2,3 has the same volume and surface area as the box with dimensions (65+-sqrt(109))/49 and 7/2, or approximately 1.113,1.540,3.5.

In 4D, I don't think I was clear enough that there are three constraints (on the 4D volume, the 3D surface volume, and the area of the 2D facets that don't have a clean analog. This leads to another 1D solution space, but this time the intuition is correct that the box with the longest edge also has the shortest edge.

[voxen]

Gah.

[madcaptenor]

In even numbers of dimensions, increasing the length of the longest edge decreases the length of the second-longest edge, increases the length of the third-longest edge, ..., decreases the length of the shortest edge.

In odd numbers of dimensions you get similar alternation, but that means that increasing the length of the longest edge increases the length of the shortest edge.

Your problem reduces to fixing the coefficients of x^0, x^1, ..., x^(d-2) in a monic polynomial of degree d and asking how changing the coefficient of x^(d-1) affects the roots. Then I wave your hands and draw some pictures, except:
1. I didn't actually draw the pictures, because I don't have a whiteboard in my shower;
2. even if I did it would be too much trouble to show them here.

[loveschak]

Correct.