This week the question is: Can You Spy on the Infinite Corridor?
You’re a senior member of the Fiddler Spy Agency, and you’ve infiltrated the enemy base, as diagrammed below. You currently find yourself standing in the middle of a narrow corridor, 1 meter wide. You are 1 meter away from a square turn in the corridor, around which is a very long “infinite corridor” (so named because, well, it’s very long.)
Importantly, there’s a flat mirror placed at a 45 degree angle in the far corner of the turn, as shown above. The mirror forms a 45-45-90 right triangle with that corner, such that its hypotenuse (i.e., the length of the mirror) is L.
For different values of L, you can “spy on” different sections of the infinite corridor. A given point in the infinite corridor can be spied upon if there is some location on the mirror that reflects light from that point to where you are standing.
What is the minimum value of L such that the mirror allows you to spy on the entire infinite corridor? (Note that this is a puzzle in two, rather than three, dimensions.)
And for extra credit:
Now suppose the flat mirror is no longer constrained to be at a 45 degree angle with the corner. That said, it must still be flush against the corner so that it forms a right triangle.
Once again, what is the minimum length of the hypotenuse L such that the mirror allows you to spy on the entire infinite corridor?
Highlight to reveal solution:
Animation. Calculation sheets 1, 2 and 3.
Referring to sheet 1, if I look from my position to the right most part of the mirror and then out into the Infinite Corridor, I am looking along the red line. Let the distance from the corner to the top most edge of the mirror be a, and let the corner be positioned at (0, 0). Then the first part of the red line passes through (-1/2, 2) and (0, a), and the second part passes through (0, a). The first line can be calculated to be y = (-4+2a)x + a. The second line (mirrored along the line y = -x + a) will be 1/(-4+2a) * x + a. Note that (-4+2a) * 1/(-4+2a) = 1. Similar calculations can be done for a blue line, hitting the left most part of the mirror. The first part of the line is y = 2/(a-1/2) * (x+a), the second part is (a-1/2)/2 * (x+a).
The area between the red line and the blue line out in the Infinite Corridor is the area I can see. Because it’s both red and blue, it’s purple. The whole corridor is purple when the red line hits (-1, 1). (See animation.) This happens when 1 = -1/(2a-4) + a, or 2a2 – 6a + 3 = 0. Sheet 2 shows this happens when a = 3/2 – √3/2 = 0.63397, approximately. Sheet 3 further shows, that this means L = √(6 – 3√3) = 0.89658, approximately.
And for extra credit:
This time I construct the red lines to hit the corner, then construct the mirror, and then figure out where the blue line hits. See animation. (Assumption: that I shouldn’t have done it the other way around, starting with the blue line.)
Then I try to eyeball when the blue line becomes flat. (I gave up trying to calculate it.) See observation.
The mirror is between (0, 0.38) and (-0.492, 0). Therefore the mirror is length 0.62.
Stuff from ommadawn.dk (Lise Andreasen) 
