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Problem of the Week 1210

Drink Me

A simple closed curve C in the plane is called "shrinkable" if all shrunk versions of C can fit into the closure of the interior of C.

Any convex curve is shrinkable.

Any star-shaped curve is shrinkable.

Find an example of a shrinkable curve that is not star-shaped.

More details:

  • A shrunk version of C is the curve obtained by choosing some shrinking factor p between 0 and 1 and using the points (px, py), where (x, y) is on C. The problem is about all shrinkings, that is, all possible choices of the shrinking factor.
  • The fitting can use any translation and rotation.
  • We allow the shrunk curve to fit inside the set consisting of the interior of C and C itself; i.e., it need not be strictly inside C.
  • A curve is "star-shaped" if there is a point in its interior such that the straight line from the point to any point on the curve lies inside the curve.
  • The "Drink Me" potion in Wonderland caused Alice to shrink to a fraction of her normal size.

Source: Dan Asimov formulated and solved this problem in 1966.

[View the solution]

8 June 2015