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

Alright 2 things to add here 1) Be sure to checkout stemerch.com for the new recommended booklist as well as STEM related apparel, the floating globe, and more! Also you'll notice another tab for 'recommended textbooks' which are good for self study, this is a common question I get as well so will keep that updated. 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.

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.

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?

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

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.

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

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

Just guessing without thinking too hard about it, but I'm going to guess it's 16 because of this: You have 4 sides, so you have 2^4 ways of using those sides - you're using each side either an odd or even number of times. A higher number of odd/even times just ends up cancelling them down to simplify into one of the basic 16 situations. To simplify, let's just look at the left and right walls. Either you're hitting both an even number of times, the left odd and right even, the left even and right odd, or both an odd number of times. Whether the combination is left 2 and right 1 or left 10 and right 9, the same point ends up being it's center point. That's the 4 dots you put at first.

I’m late to the show, and there’s a lot of cheeky comments so far. I just want to say that I was really surprised, and really happy with the result! Thank you for this.

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

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

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

If American students paid attention in math class they'd know how to avoid shooters.

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

I would see myself probably dunno ngl

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

I love CuriosityStream! I've been subscribed for years now.

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

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

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).

A guy named Nils Berglund has an awesome demonstration of this in action with an expanding circle of dots where the 16 blockers end up creating a bunch of various sized, moving arc segments with holes. Fun to watch as every single one gets close but never actually touch the target dot. I saw that he linked to this video so it was awesome following that up with an explanation of what I had just watched!

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

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

In a room with mirrors on walls, for each pair (light source, observer) there is a set of 16 columns that will obstruct not only the light from the source, but also the light from all reflections. Now I want to see how the position of those 16 points changes as the positions of the light source and observer change.

Late to the party, but does this apply to rooms with differing number of sides? Would an octoganal room require 64 blockers? Triangular 9?

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

“to answer that, we need to talk about parallel universes”

This is so relateable

this needs to be a room in the center of a laser tag arena

The solution was surprising and surprisingly satisfying.

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

my brain just went before any explaination "just use 6 blocker around you (make sure they touch each other) and you'd be good to go" and then you used almost 3 times as many

I think a animation of the shooter and target ger moving, with the blockers responding would be cool

"Switching to your pistol is faster than Reloading."

consider: put a point directly on the shooter so no lasers can be shot, they would be absorbed immediately for infinitely many paths

the first thing i thought was hiding in the corner and adding 2 blockers to block the 2 sides

This is such a cool explainer. Incredible work if this is originally yours

Anyone who has seen the DVD logo bounce on a screen knows the safest spot is the corner

Yeet the block at the shooter when he's distracted.

I am just realizing that this is the same guy who makes all the comedy sketches that I watch. I had no clue about this, and am doubting life as I know it. I never thought of him as a mathematician.

I didn't know you were a learned man as well!

so then if set up properly, the target and the shooter wouldn’t be able to see each other in the mirrors

This was a really cool puzzle. I didn’t entirely understand it but it was cool.

I've thought about this ever since watching the dvd player screen saver, waiting to see if it can ever hit perfectly in the corner of the screen.

Before watching, I have 3 ideas for an amount to protect myself. 1: if blockers can be any shape of any size: 1. Just a wall through the whole room. 2: if blockers can be rectangles of limited size, 3. Triangular around me, trapping me. 2: if they have to be circles of equal size to my footprint circle, 6. Hexagonally around me. Touching me as well as themselves, leaving no gap.

This is definitely relatable. The other day I was stuck inside a rectangular room with the four walls made of mirrors. And there was someone trying to shoot me with a laser gun! What a horrifying experience! Fortunately I was able to locate the 16 blockers on the correct coordinates. So I survived

"Can't we just do this?" "No we can't." "Ok, but we could if we turned this problem into an entirely different one."

so relatable man

I’m in algebra 2 rn and we have been talking about i and this reminds me of it because after just 4 it repeats itself

I feel like this could work well as a puzzle video game. You have 16 blockers at the beginning, and a very unskilled opponent, who fires directly at you with one laser, but as you get further into the puzzles, the opponent uses more lasers, takes unusual paths, and you get less blockers over time.

Ah, yes, now I just need to prepare 16 blockers to be carried with me everywhere I go in case I land in this super relatable situation again.

This situation is so normal I’m starting to hate it but it does make me feel smart

Crazy how 16 one CM blockers block ALL paths

Getting this comment a lot so just want to emphasize that these blockers are NOT pixels, they are infinitely small (zero dimensional) points. So no you cannot surround the shooter or target with 8 of these blockers (or any finite number), to actually surround one of them you'd need an uncountably infinite number of blockers.

This feels like the kind of thing that would show up in an anime or sci-fi movie or something. "Why can't my lasers hit you!?" "Well you see..."

Gotta say, knowing Zach only from his skits and not checking the channel name when clicking the video, seeing him pop up was an absolute shock lmao

8. Make a 3x3 with you in the middle, assuming both blockers and target are of the same circumference. You can also make a circle of blockers with the same number of blockers.

Before I start watching: Judging by the experience of looking at DVD logos, stay in the corners. After I watched the video: Ah, you mean setting up the blockers, not positioning yourself.

I’ve played enough laser tag with mirrors to know that as long as you can see them in the mirror you can shoot them. So you would just need enough blocks to block off line of sight. Simplified.

Cool. I really liked this.

Can the number of blockers change depending on the format of the room? It probably isnt the same if the room is a triangle

Me, a geniuos: Simply place the blocker on the shooter

Having played a lot of Portal 2 community maps, I really can relate to this problem.

This is cool stuff. If you had controlled force field. You could save a bunch of energy with only 16 points instead of a sphere.

Styro pyro needs to do this

I just realized that the shooter also wouldn’t be able to see you

Me: puts a blocker directly at the shooter *"Sometimes my genius is almost frightening"*

A Cat In That Room Would Suffer From Not Being Able To Move

Me: just surrounds myself with blockers that slightly overlap

I know nothing about math so all of this sounded like magic but I felt like I learnt something so I have that going for me. Cool video 😎

So, if we are in two dimensions, simply stay in a corner, though, I know for this scenario we are trying to accomplish the task of protecting no matter where the two shooters are(assuming they aren’t sitting next to each other or, as mentioned, the recipient is in the corner. Infinity is nuts on a 2 dimensional plane. Go crazy and do 3 and 4 dimensions.

Not me thinking about making a 3x3 square around myself

That card that said “the globe you are staring at” actually caught me off guard

1:32 I was just thinking "woah what's that cool spinning thing in the background" and then the i-card appeared

“ The extremely relatable situation” Yea ok I didn’t know that

Me, an intellectual: puts the blocker infront of the shooter

"We'll work up to infinity" I don't know man, I haven't got all day, or the lifetime of the universe for that matter

Go to a corner and then all you have to do is put a blocker around you so does that mean that your safe? Also it depends on the size of the blocker because if it’s the size of a atom will the laser just go round it? And it depends on the size of the laser because if it’s the size of the room then it will just hit you if there is the tiniest gap in your defence

This would make a fun flash game

This is so interesting. Math used like this expands the mind I believe.

You're probably not reading this, but could you also do it for 3 dimentional space ?

Watching the set up, it immediately reminded me of a scene in the anime Zatch Bell, where it was this same exact scenario of being in a room of reflective surfaces, and the protagonist figured out that they can predict to stand in specific spots to avoid a laser shot by his opponent. Looked it up, it’s the fight in episode 16 of season 1.

3 blockers if you place them good enough, or tou can troll and put block around the shooter, or you can be a rebek and break a mirror, and bring down the amchine

That is an incredible puzzle. At first I was convinced the answer would have to be uncountably infinite, then after seeing the first step of the proof I thought it would be countably infinite, as we can represent each possibility on a NxN grid. And then the pattern starts repeating, very cool stuff!

It'd be interesting to do this in real life. Have a room with mirror walls, floors, and ceiling. Add a small light source, and 16 pillars a bit thicker than the light source covered in vantablack, black 3.0, or another really dark substance. Mark a very specific place where someone can stand (in dark clothing), close one eye, and see no light. It would be so cool opening the other eye (or just moving a bit) and seeing the light. The only flaw I see is how flat we can make mirrors, but it seems doable.

someone needs to make this into a game

if you are in a randomly chosen spot and you end up in the corner then the smallest amount of blockers would be 2 surrounding you

Stand at any point in the square room, and look around yourself. Anywhere where you can see the shooter, either directly, through a mirror, or through multiple mirrors, put a blocker in between. Once you cannot see the shooter anymore (all places where you previously could see the shooter are now blocked), you are safe.

This is actually very similar to a coding interview I got from Google! The difference was that the laser did lose energy and you had to count all of the ways the laser hit the target, and the shooter was a target. I still landed on the reflection strategy, tho. First instinct: countably infinite, were gonna talk about Ulam's spiral and counting rational fractions

also are we counting the entire ball/blocker or just the center

When you draw a circle in a plane of radius 1 you can perfectly surround it with 6 other circles. so you could do all that using 16 blockers or use 6 of the same radius to block block it, by surrounding the point of origin with circles of the same radius.

Starting from today I'm gonna always take 16 bricks with me! You never know...

Now i finally see the purpose of Raytracing, somebody needs to whip out their RTX and make this into a simulation.

This video has significantly enhanced my billiards playing skills

What if you don't place the blockers in the middle of the path? Will you get more value or less value

Me, an intellectual: *just stand in the corner and cover yourself with 3 blockers lmao*

That was a major call-out to me staring at the globe lol

Here is a (greatly altered) version of the problem. Let's assume that there are no blockers, but the shooter can place static mirrors before shooting the laser. If the laser's path will not be affected by hitting the target, is it possible that the shooter does not shoot themselves?

Thanks! Now, I know what to do if I'm stuck in a room of mirrors and someone's trying to shoot me with a laser!

What would happen in different shaped rooms? (Triangle, hexagon...) Perhaps even 3 dimensional rooms, like say a cube, I wonder what you would get as an answer then!