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538 Riddler: Puzzle of the Overflowing Martini

19 May ’16

Here's my solution to this week's 538 Riddler:

If the liquid reaches fraction of the way up the glass when upright, then the liquid goes fraction of the way up the glass on the opposite side just before it begins to pour.

Dandelin spheres were key to my proof. They prove that the top of the liquid (a conic section) forms an ellipse. They also prove that a sphere tangent to the top of the liquid and the sides of the glass intersects that elliptical conic section and one of its foci! I derived two expressions for the volume of the liquid (one pouring, one upright) and set them equal to one another. Interestingly, the steepness of the glass ( in my proof), drops out of the expression!