Could you avoid being hit by a laser if you were in a room of mirrors?

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[zachstar]

Alright 2 things to add here

2) Spoilers down below but this is where I want to acknowledge what I mention at the end of the video. I never explained WHY there is repetition after you reflect the midpoint back to the main room and the best explanation I got is quite a mouthful and it involves modular arithmetic. Let's say that the target is at point (x, y) (and we're just going to consider the horizontal reflections). After you reflect it to the right, the reflection lands at (2-x, y) and after another reflection it goes to (2+x, y) then (4-x, y) then (4+x, y) and so on. As you can see, reflected points all land at (2n +- x, y) (as in the x coordinates are just even numbers plus or minus x). 

Adding in some modular arithmetic you'll notice these are all congruent to x (mod 1), or -x (mod 1) . That's all that can happen as you reflect a point about the left or ride side of those 1x1 squares actually, the point either stays the same (mod 1), or it becomes negative (mod 1).

Now again, all the x coordinates of the reflected targets can be written as 2n +- x, meaning the midpoints would be (2n+x+u)/2 = n + x/2 + u/2 and (2n-x+u)/2 = n - x/2 + u/2 (assuming the shooter has coordinates (u, v)). So we have two sets of midpoints and these midpoints go on forever with n. When those points are reflected back, as we've seen, they either stay the same mod 1, or become negative mod 1. So we have 4 different results, n+x/2+u/2, n-x/2+u/2, -n-x/2-u/2, and -n+x/2-u/2 (all mod 1). This seems like infinitely many points still because of n, but n can be dropped from all of these because it doesn't change the value mod 1. For example 1+x/2+u/2 = 2+x/2+u/2 mod 1, meaning they are the same point in the original 1x1 room. So we have 4 answers, x/2+u/2, -x/2+u/2, -x/2-u/2, and x/2-u/2 (all mod 1), these are all the x coordinates after the midpoints are reflected back, there can be nothing more. So that's 4 different x coordinates and the same thing can be done for the y coordinates, leaving us with 16 points in total. 

Then to finally to answer the other question of how can you be safe with less than 16 blockers, it can happen if the x (or y) coordinates of the shooter and target add to 1. Because then x/2+u/2 and -x/2 - u/2 are now congruent mod 1, and the other two are also congruent. So what was 4 different x coordinates becomes 2.

[Julian-ijzm]

Man i hate when im in a perfectly square room made of mirrors and there is someone trying to shoot me with a laser which will never lose its energy as it reflects, thank you for the video.

[lordheaviside]

To all people who say math has no applications, what a fool you must feel now! Who can say he has never been in this extremely relatable situation?

[mt]

Now we just need someone to individually animate all the laser paths

[ShaeTollefson]

This has happened to me 7 times now and I’ve died every single time. Thanks for the tip for when it happens again!

[frgal]

*Me when I saw the title:* oh well you just wait until the light dissipates.
*the video immediately:* It never loses its energy.

[tonycatlett]

Mathematicians: Uses this information for intellectual purposes
Me: uses this information to beat my friends in air hockey

[quietstories]

Room of mirrors where a lazer that never runs out of energy trying to kill you sounds like an SCP entry

[phillipgan]

interesting stuff. i'd love to see some sort of interactive demo where you could drag around the two points and see how all the blockers would have to move to compensate.

[jigyasbaruah]

Damn now I know what would happen if I was stuck in a mirror room with lasers

[cononsberg]

Pretty cool how, in a room full of mirrors, this would also completely hide you from a target.

[jayxi]

Just say no and the light won't touch you because it isn't legally allowed touch you without your consent. No need of those fancy 16 blockers

[Pax.Britannica]

Going from comedy sketches to in-depth presentations on something as interesting as this, is a pretty amazing transition, GG

[ChannelJeffrey]

The solution was surprising and surprisingly satisfying.

[baconking]

the 20 dislikes are from people who surrounded themselves with 8 points thinking they were circles.

[CharlesB]

This use of reflections is also one method for calculating bank and kick shots in pool off of one or multiple rails (just factor in a little bit of physics about speed and friction).

[giovannirozatti]

If DVD screens taught ne something: yes if you stay on corners.

[ThingEngineer]

Kid, “math is so stupid why do we have to learn it, I’m never going to use it?”
“Yes you will, because lasers.”

[noahgranger]

Math puzzles:
"We'll start small and work up to infinity"

[ant-mfkl]

The amount of peope saying "just put 4/5/6 blockers around you" is genuinely worrying