Grades 11-12 Video Solutions 2013-Levels 11&12 Video Solutions 2013 problem13

Levels 11&12 Video Solutions 2013 problem13

Video Transcription

Question number 13.  A circular rug is placed on the floor of square tiles. All the tiles which have more than one point in common  with the rug are marked gray. Which of the following is an impossible outcome?  What I'm going to do here in terms of a solution is draw circles to decide  which situation is possible and which is not.  For a more concise mathematical solution, you can consult the written suggested solutions.  So let's begin by looking at the easiest situation and see if I place the center of the rug at the center of the central tile and draw something like this.  We see that the center tile is completely covered up and the remaining tiles share at least two points with the rug.  We can do a similar thing in B by finding the center and drawing a circle that's tangent to the edges.  In B, it gets slightly more difficult. The outstanding tile to the right causes problems, but if we draw a circle that is passing  through the corners here like that, those are two intersections with the tile on the right and so we get to include that tile.  And that gives us an idea on how to place the rug in A. What we need to do is circumscribe the central tile.  Like so. So the corners are included, but those are the only points in common.  And essentially we have exhausted our possibilities. Now just for completeness, let's look at E and draw a circle  that includes the corners of the two missing white tiles. If we do that, we see that the circle is covering up the four central squares.  However, the tiles marked here with an X have only one point in common with the circle.  And so that is the impossible situation and the answer here would be E.

Video Summary

The problem involves determining how a circular rug can cover square tiles and identifying an impossible arrangement. By analyzing different placements, one can see that the tiles must share at least two points with the rug to be marked gray. In options A and B, circles can be drawn to include more than one intersection point with respective tiles. However, in option E, some tiles are marked with only one common point with the circle, making this an impossible outcome. Thus, the solution concludes with option E as the impossible scenario.