Here are four problems from the puzzle column in the latest newsletter from MSRI, faintly edited. —Dan ----- 1. Find all possible positive integers a, b, c, and d such that ab = c+d and cd = a+b. 2. Let C be the surface of a cube in 3-space. What is the largest n such that there is a planar regular n-gon P, not lying entirely on one face of C, such that all P’s vertices are on C? 3. Suppose there are n hats, each with a different color. These hats are placed on the heads of n sages. All of the sages know all the colors: their own hat and everyone else’s hats. A referee then announces the "correct" hat color that should be on the head of each sage. The sages are then allowed "swap" sessions: in one session disjoint pairs of sages are allowed to interchange their hats. Can the sages fully correct their hat colors in two swap sessions? 6. N people are randomly placed on a circle. Each person turns to face their closest neighbor (with probability 1, all the distances are distinct, so this is sufficiently well-defined). What is the expected number of "lonely people" who are looking at a neighbor who is not looking at them? -----