#16. Covering a Cube Minus One Vertex

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Prompts for discussion:

1. We saw during the discussion two different ways of covering the cube minus the origin with (d) hyperplanes: we can use the hyperplanes \left\{x_i=1\right\}, i=1,2, \ldots, d, or the hyperplanes \left\{x_1+x_2+\cdots+x_d=k\right\}, k=1,2, \ldots, d. Are there other ways? Can all the ways be counted?

2. What happens if you are trying to avoid two points? Note that it now perhaps matters which two!