Can two cuboids with different side lengths have the same volume and perimeter? We know that two rectangles with different side lengths cannot have the same area provided their perimeter is the same. But two cuboids with different side lengths can (as evidenced by the example 1,6,6; 2,2,9 in an answer below) have the same volume and perimeter. Side lengths are constrained to be integers.
I have glossed over it a bit; but can't seem to find a proof.
 A: How about $1, 6, 6$ and $2, 2, 9$?
A: I can generalize paw88789's example. For all $n>1$ the formula $(1,2n^2-n,2n^2-n)$ and $(n,n,(2n-1)^2)$ produces two different cuboids that have the same total edge length and volume.
I can also show why this can be done in three dimensions and not two. Let $l$ be the rectangle's length. Let $w$ be the rectangle's width. Let $P$ be a known value the rectangle's perimeter. Let $A$ be another known value the rectangle's Area. Therefore the two following formulas describe the rectangle. $2l+2w=P$ and $lw=A$ There are two equations and two unknowns therefore rectangles can be uniquely identified by their perimeter and area. Applying the same procedure for the cuboids, let $h$ be the height of the cuboid. We get two formulas $4l+4w+4h=P$ and $lwh=A$. This is two equations and three unknowns, so knowing a cuboid's total edge length and volume doesn't uniquely identify the cuboid. It then becomes possible to construct two different cuboids with the same total edge length and volume.
