- Let P be a simple polygon with n>3 sides. A simple polygon is a polygon that does not self-intersect, but it is not necessarily convex. Prove that no matter the shape of P, there is always a diagonal (a segment that connects two vertices of P without intersecting any of its sides) that divides P into two polygons, both of which have at least n/3 sides.
- Let A and B be two points in the plane. Using only a compass and a straightedge, find the point C which is the exact middle point between A and B. Now do the same thing, but
*using only a compass*. - There are 17 point-sized light-houses in the plane. Assume that each of these lighthouses can shine light in any direction with an angle of 2*pi/17. Prove that no matter the position of each lighthouse, it is always possible to choose the angles at which they shine their light such that every point in the plane is illuminated (point-sized lighthouses don’t cast shadows).

In Selective Enhancement of Specific Capacities Through Psychedelic Training, Willis Harman and James Fadiman outline the results of a study about the potential use of psychedelics for problem solving. In the study, scientists, engineers, mathematicians, and designers took either 100 micrograms of LSD or 200mg of mescaline and worked on a problem they were personally invested in and which they had not been able to solve for at least 3 months. According to Fadiman, 9 out of 10 participants came up with a solution to the problem that was validated by the participant’s professional colleagues.

The three problems above are not easy, but they are also not insanely difficult. If it means anything to you, their level of difficulty might be around that of a problem 1 or 4 of an IMO, with the advantage that you do not need any fancy math to solve them (high-school math is more than sufficient). I do not know if solving these problems is easier or harder on psychedelics, but I figure I would share them as possible Schelling points for “challenging math problems to think about while on psychedelics” to see if anyone reports benefits from such a setup. I personally like these problems, and I can assure you that they do have interesting and clever solutions. Assuming you are already planning on taking a psychedelic substance in the future: I would recommend trying to solve one of these problems for at least 1 hour while sober, and then setting aside at least 30 minutes (preferably 1 hour) while on a psychedelic and giving it your full attention. Please let me know if you either solve the problem or get an interesting insight from such an exercise. I am particularly curious to hear about *what aspects* of the psychedelic state seemed to be either beneficial or detrimental in solving these problems. Even if you do not solve the problem, you may be able to think about it in new ways and derive useful insights. Again, if you do so, let me know as well.