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)?

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

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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?

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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?

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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?

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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?

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Is the crossing number additive when taking knot sums?

 

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

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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?

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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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