Imagine that $k$ runners on a $1$ mile long circular running track start running with different constant speeds. Pick one of the runners. Will there be a time when this runner is $1/k$ miles away from all other runners?
Does there exist a rectangular box such that all of its side lengths, all of the lengths of the diagonals of the faces, andthe length of the long diagonals, are all whole numbers (integers)?
Is there a rectangle built out of some number of smaller rectangles, so that all the smaller rectangles have equal areas but different perimeters, and so that all the side lengths of all the rectangles are whole numbers?
Paint an infinite flat wall so that whenever two points are 1 meter apart, they are painted different colors. What's the least number of colors you need to do this?
Consider a convex polyhedron in which every face has an even number of edges and every vertex has three faces around it. Can you go for a walk along the edges, coming back to where you started, and passing through each vertex exactly once?
Build a polyhedron by taking lots of 1X1X1 cubes and gluing them together face-to-face in some interesting way. Can you unfold this polyhedron by cutting along edges to get a single flat shape with no overlaps?
