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Three runners

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?

Twin primes

Are there are infinitely many primes p such that p + 2 is also prime?

Seven rectangles

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?

Chromatic number of the plane

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?

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?

Is the crossing number additive when taking knot sums?

Path on a cube

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?

Does every polyhedron have a periodic orbit?

Is there an amphichiral 2-component link with linking number 8?

Pascal's Triangle

Does any number appear exactly 5 times in Pascal's triangle?

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