Grades 7-8 Video Solutions 2022-2022_7-8

Video Transcription

Question 26. What is the smallest number of cells that need to be colored in a 5x5 square so that any 1x4 or 4x1 rectangle laying inside the square has at least one cell colored?  So if we draw a 5x5 grid, one of our effective ways to fill out the cells is like so.  This keeps track of any of the cells in the first three columns and the top three rows, and makes sure there is always one black cell for a 1x4 or a 4x1 shape, except in one instance.  So if we mirror this on the other side, we are guaranteeing no 1x4 or 4x1 squares.  If we removed one to try to get answer A of 5, we can see that it will be possible for a 1x4 square to appear. So, our answer will be B, 6.

Video Summary

The smallest number of cells that need to be colored in a 5x5 grid to ensure every 1x4 or 4x1 rectangle within it contains at least one colored cell is six. By strategically coloring cells in certain columns and rows, you can guarantee coverage for any 1x4 or 4x1 rectangle. Attempting to color only five cells allows for the possibility of missing coverage for some rectangles. Therefore, having six colored cells is the minimal and effective solution to cover all possible 1x4 or 4x1 rectangles.

Keywords

5x5 grid

1x4 rectangle

4x1 rectangle

colored cells

minimal coverage