Beautiful Geometry

The irregular purple polygon above, made of five congruent squares, is the floor plan of an art gallery.

Your job is to position some number of unmoving guards — who cannot see through walls — such that every location in the gallery is in view of at least one of the guards. It's possible, as shown in the example, to guard this particular museum with two guards.

Is it possible to guard this entire gallery with only one guard?

Guards in the Gallery

           

Using the same rules as before, what's the fewest number of guards needed to guard this gallery?

Guards in the Gallery

           

This particular gallery is a little more irregular and isn't just a set of squares joined together.

Using the same rules as before, what's the fewest number of guards needed to guard this gallery?

Guards in the Gallery

           

Using the same rules as before, what's the fewest number of guards needed to guard this gallery?

Guards in the Gallery

           

One last problem, and the trickiest of the set! What's the fewest number of guards needed for this gallery? (You can assume any region of the gallery that appears to be a rectangle is, in fact, a rectangle.)

Guards in the Gallery

           

You might start to suspect there is a systematic way to solve this kind of problems, and there is!

As part of the course, we'll teach a truly wonderous coloring proof for finding the fewest number of guards needed for this kind of puzzle, and look at some twists like "internal walls" and "worst-case scenarios".

Proceed onward to learn some beautiful geometry!

Guards in the Gallery

           
×

Problem Loading...

Note Loading...

Set Loading...