Question 2
For a positive integer $m$, consider an $m\times2018$ grid with some cells blank and the others containing 0 or 1. Call the grid complete if every binary string $S$ of length 2018 can be obtained by filling the blank cells in some row with zeros and ones. A row already equal to $S$ also counts. Call the grid minimal if it is complete but ceases to be complete after any row is removed.
(a) For every integer $0\le k\le2018$, prove that a minimal $2^k\times2018$ grid exists with exactly $k$ columns containing both 0 and 1.
(b) If a minimal $m\times2018$ grid has exactly $k$ columns containing both 0 and 1, prove that $m\le2^k$.
A binary string of length 2018 is $x_1x_2\cdots x_{2018}$ with each $x_i\in\{0,1\}$.
(a) For every integer $0\le k\le2018$, prove that a minimal $2^k\times2018$ grid exists with exactly $k$ columns containing both 0 and 1.
(b) If a minimal $m\times2018$ grid has exactly $k$ columns containing both 0 and 1, prove that $m\le2^k$.
A binary string of length 2018 is $x_1x_2\cdots x_{2018}$ with each $x_i\in\{0,1\}$.