A Game Of Circles

Alice has two circles on the plane that intersect each other twice. Bob wants to know these circles. To do that, Alice plays a game with Bob. Alice selects \(n\) points from each circle in such a way that both intersections are selected (so Alice selects \(2n-2\) points). Bob then asks for the identities of \(k\) of them.

What is the smallest value of \(k\) required so that Bob can always determine the circles, even if Alice tries to prevent it? Enter your answer for the following three problems:

  1. \(n = 6\)
  2. \(n = 5\)
  3. \(n = 4\)

If Bob cannot figure out the circles with any \(k\), enter 0. Otherwise, enter the minimum value of \(k\) such that Bob can always determine the circles. Concatenate your answers for all three problems. For example, if the answers are 0, 10, 100 in that order, enter \(010100 = 10100\).


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