[math-fun] Two random intersection puzzles

1. Given a convex quadrilateral, consider the four bounding line segments (i.e. the sides) and the two segments obtained by joining the midpoints of opposite sides — six line segments in all. Let L be a line not parallel to any of the six line segments. Show that a random translate of L, conditioned to intersect at least one of the six segments, can be expected to intersect 3 of them on average. 2. Consider the twelve line segments in a planar projection of the 1-skeleton of a parallelepiped. Let L be a line not parallel to any of the twelve line segments. Show that a random translate of L, conditioned to intersect at least one of the twelve segments, can be expected to intersect 4 of them on average. Jim Propp