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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?

Perfect numbers

Does an odd perfect number exist?

Non-invertible knot

Find a method to determine whether or not any given knot is invertible.

Triangulating a donut

What is the best way to cut a donut up into tetrahedra?

Can you cut a rectangle into 3 congruent non-rectangular pieces?

n/2 and 3n+1

Pick a positive whole number n; if odd, replace with 3n+1, if even divide by 2, repeat ...

Perfect numbers

Are there infinitely many even perfect numbers?

Penrose tiling

Is there a single aperiodic tile?

Does every triangle have a periodic orbit?

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