I'm having trouble even beginning to figure out how to approach solutions for this. I begin with a unit cube, and imagine all the possible lines that intersect the cube. I am assuming there must be an average length of these intersections; I want to find that average length. Another way to express it is, for every point on and within the unit cube, what is the average length of all possible lines through those points, length meaning on and within the cube? Here is where I am so far... There is a group that intersect the cube through one perimeter point: - those that pass through one of the vertices - those that pass through one point of an edge There is a group that intersect the cube through two perimeter points: - those that enter and exit through adjacent vertices and intersect all points of the connecting edge (12 of these) - those that enter and exit through opposite vertices of a face and intersect all points of a face diagonal (24 of these of length SQRT(2) - those that enter and exit adjacent edges of a face and pass entirely through the face - those that enter and exit opposite edges of a face and pass entirely through the face - those that enter and exit through opposite faces - those that enter and exit through adjacent faces - those that enter and exit through opposite vertices of the cube - 4 of these and they are the longest at SQRT(3) I think that is all...? It is not clear to me how to distinguish some instances of some groups - when or whether to include vertices as parts of edges, or when or whether to include edges as parts of faces. Is there a convention for this? I know there are infinitely many intersection lengths here, but the shortest are zero where the intersection is a point, and the longest are SQRT(3) between most distal vertices. For a while I was wondering if it would be mathematically clearer to use a unit sphere rather than a unit cube, but I have no proof the average intersection length would be the same... but that just got me wondering what radius sphere would have the same average intersection length as the unit cube. Maybe I should back up a dimension and work with the average intersection length through a unit square? The average intersection length through a point looks like zero. The average intersection length through a unit line looks like zero, too; one instance of length 1 and an infinite number of length zero The average intersection length through a unit square looks like it is greater than zero and is less than SQRT(2). I'm thinking that using calculus on trigonometric functions is the way forward, but it looks like the problem may need to be broken up into parts corresponding to the different groups of intersections? Anyone have any clues to figuring out an approach (maybe to just the unit square for now)?