Consider the following entry from my notebook (16-Feb-2014):
The Art Gallery Problem: An art gallery has the shape of a simple n-gon. Find the minimum number of watchmen needed to survey the building, no matter how complicated its shape. [Source: problem 25, chapter 2, Problem Solving Strategies, Arthur Engel]
Hint: Use triangulation and colouring. Not an easy problem, and in fact there is a book dedicated to the theme of this problem: Art Gallery Theorems and Algorithms by Joseph O’Rourke (see chapter one for detailed solution). No reflection involved.
Then we have a bit harder problem when we allow reflection (28-Feb-2017, Numberphile – Prof. Howard Masur):
The Illumination Problem: Can any room (need not be a polygon) with mirrored walls be always illuminated by a single point light source, allowing for the repeated reflection of light off the mirrored walls?
This reminds me of the much simpler theorem from my notebook (13-Jan-2014):
The Carpets Theorem: Suppose that the floor of a room is completely covered by a collection of non-overlapping carpets. If we move one of the carpets, then the overlapping area is equal to the uncovered area of the floor. [Source: §2.6, Mathematical Olympiad Treasures, Titu Andreescu & Bogdan Enescu]
Why I mentioned this theorem? The animation of Numberphile video reminded me of carpets covering the floor.
And following is the problem which motivated me write this blog post (17-May-2018, PBS Infinite Series – Tai-Danae):
Secure Polygon Problem: Consider a n-gon with mirrored walls, with two points: a source point S and a target point T. If it is possible to place a third point B in the polygon such that any ray from the source S passes through this point B before hitting the target T, then the polygon is said to be secure. Is square a secure polygon?