Question 1192533
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ANSWER: 7<br>
Obviously, if any row or column contains no colored cells, then you can place a 1x4 or 4x1 rectangle in that row or column without covering any colored cells.  So to start with, you need at least 5 of the cells to be colored, with one cell in each column and one cell in each row colored.  There are many ways you can do that.  As examples....<br><pre>

   +---+---+---+---+---+   +---+---+---+---+---+   +---+---+---+---+---+
   | X |   |   |   |   |   |   | X |   |   |   |   |   |   | X |   |   |
   +---+---+---+---+---+   +---+---+---+---+---+   +---+---+---+---+---+
   |   | X |   |   |   |   | X |   |   |   |   |   | X |   |   |   |   |
   +---+---+---+---+---+   +---+---+---+---+---+   +---+---+---+---+---+
   |   |   | X |   |   |   |   |   | X |   |   |   |   |   |   | X |   |
   +---+---+---+---+---+   +---+---+---+---+---+   +---+---+---+---+---+
   |   |   |   | X |   |   |   |   |   |   | X |   |   |   |   |   | X |
   +---+---+---+---+---+   +---+---+---+---+---+   +---+---+---+---+---+
   |   |   |   |   | X |   |   |   |   | X |   |   |   | X |   |   |   |
   +---+---+---+---+---+   +---+---+---+---+---+   +---+---+---+---+---+</pre>
Whatever way you choose those first five cells to be colored, there are four colored cells that are either in row 1 or 5 or in column 1 or 5.  That means there are four places you can place a 1x4 or 4x1 rectangle without covering a colored square -- two columns and two rows.<br>
By placing another colored cell at each of the two intersections of those two rows and those two columns, you eliminate all the places where a 1x4 or 4x1 rectangle could have been placed without covering a colored cell.  Using the above examples....<br><pre>

   +---+---+---+---+---+   +---+---+---+---+---+   +---+---+---+---+---+
   | X |   |   |   | Y |   |   | X |   |   |   |   |   |   | X |   |   |
   +---+---+---+---+---+   +---+---+---+---+---+   +---+---+---+---+---+
   |   | X |   |   |   |   | X |   |   | Y |   |   | X |   |   |   | Y |
   +---+---+---+---+---+   +---+---+---+---+---+   +---+---+---+---+---+
   |   |   | X |   |   |   |   |   | X |   |   |   |   |   |   | X |   |
   +---+---+---+---+---+   +---+---+---+---+---+   +---+---+---+---+---+
   |   |   |   | X |   |   |   | Y |   |   | X |   |   |   | Y |   | X |
   +---+---+---+---+---+   +---+---+---+---+---+   +---+---+---+---+---+
   | Y |   |   |   | X |   |   |   |   | X |   |   |   | X |   |   |   |
   +---+---+---+---+---+   +---+---+---+---+---+   +---+---+---+---+---+</pre>
So the smallest number of colored cells that make any 1x4 or 4x1 rectangle cover at least one colored cell is 5+2=7.<br>