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U N S O L V E D :

Does there exist a rectangular box such that all of its side lengths, all of the lengths of the diagonals of the faces, and the length of the long diagonals, are all whole numbers (integers)?

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?

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?

Is the crossing number additive when taking knot sums?

Does every polyhedron have a periodic orbit?

A hypothetical situation

Is there a square with integer side lengths and a point in the plane whose distance to each corner of the square is an integer?

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U N S O L V E D :

Imagine that $k$ runners on a $1$ mile long circular running track startrunning with different constant speeds. Pick one of the runners. Willthere be a time when this runner is $1/k$ miles away from all otherrunners?

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

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?

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?

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?

Is the crossing number additive when taking knot sums?

Does every polyhedron have a periodic orbit?

Is there a square with integer side lengths and a point in the plane whose distance to each corner of the square is an integer?

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?