Triangle hole trick: made 3D from cubes and explained

Also known as the "missing square puzzle," this great little trick is sure to get a reaction at parties- especially from those who haven't had time to think about it before! If you are going to build one, I recommend the cubes over the flat parts: the cubic version is easier to handle, more convincing, and more forgiving. This video first outlines some of the construction process, and then provides a conceptual explanation for what's happening here. Spoiler: there are three triangles, and all three are slightly different- the video explains how.
The overall puzzle has an area of ½bh, or ½(13)(5), or 32.5 units^2. The sum of the areas from the four parts is still one unit less, so something is amiss. There is a slight variation to the shapes of the three triangles here, but that is not easily visible to us, especially since we have a natural tendency to quickly make assumptions about the things that we see. I'm not criticizing; sure, this tendency can cause problems, but it's also the source of some great advantage to us. Because we can make assumptions, we can learn from both the world and others readily, which lets us quickly adapt to changing circumstances. There will always be those who consider this trusting trait as gullibility, and as a thing to be exploited, but that is only because dishonesty is so rampant. If it weren't for those selfish fools who are endlessly attempting to swindle the honest with some new snake oil, humanity would suffer much less from lost productivity. Your greed and aims for personal success are the result of a short-sighted, self-oriented evolutionary origin that you should really consider suspect. Our little, harmless personal biases can have lasting social reverberations, which reflect themselves in our so many failed attempts in the political and economic sphere(s) throughout history.
Slight numerical differences can have amplifying ramifications as a system becomes larger and more complex; many of these emergent effects are not immediately apparent at the outset, given the small values in differing orientation to the seed variables (the initial conditions), which may splinter and then crystallize into trees of branching potentialities. This is another one of those butterfly effect things, but the point is that details matter. In a more real-life example, using a six-inch pocket level to make sure that an eight-foot bench is level can be disastrous if you're off a bit: there is a total of 16 lengths of 'pocket level' within the length of your bench, so your error will be multiplied by that factor. Thus, ⅛" out of level becomes two inches on your bench. Little mistakes, err, or, error- can be projected; error becomes amplified in affect as time, space, frequency, use, or any other dimensionality is added into the parameters of a given system.
Rabbit hole:
• the triangle puzzle's ...
Music:
"Desert City" Kevin MacLeod (incompetech.com)
Licensed under Creative Commons: By Attribution 3.0 License
creativecommons.org/licenses/b...

[@smalls835]

first!

[@pocket83]

I confer upon you this badge: the prestigious *_pinned heart._*

[@alexwang982]

Smalls lol

[@nickjimenez9983]

No one cares

[@diamondhawk7427]

Nick Jimenez the creator of the video clearly does

[@Notlf]

Diamond Hawk74 savage

[@MarCel-ih6ui]

That, I say, is truly the type of quality content that KZbin is lacking. Thanks.

[@pocket83]

That's one of the best comments I've received in a while. I graciously and humbly agree. Thank you.

[@bno112300]

If I ever become internet famous, do you have any reservations about sending people to your channel? You seemed concerned with floods of dimwits in the past. I, on the other hand, enjoyed your video game content.

[@pocket83]

No reservations at all. I appreciate that! In fact, this channel's pretty old now, so I've already been passed up a number of times by some of the early commenters. I'm too much of a smart mouth in the comments, and too little sensationalistic on video, so I expect that you can out-famous me if you try. Be sure to thank them for watching, as if they are entitled to your content and they are graciously bestowing upon you their view(s). I did that early on, and I grew quickly. Once you're big, just come back once in a while to pity me for my obstinate refusal to pander the masses; quietly scoff at me, like the young scholars did at old Diogenes as they passed by his wine-cask apartment. Or, we could do some collab gaming videos. "Game Cynics." I like it.

[@yungjose3369]

Zu'u Agos there is no quality content youtube is lacking. The thing is the entertainment it gives to different audiences.

[@4thgradedropout980]

So a better never ending chocolate bar? Very cool.

[@pocket83]

Yeah, I remember that one. Very similar. I think it even uses the same dimensions if I recall correctly.

[@4thgradedropout980]

pocket83 Perhaps. Cheers for more quality content.

[@Dan-om4no]

i came here to learn about a puzzle mystery and i also learned how to cut squares lmao quality vid

[@evanbaker10]

its like the chocolate trick

[@Kintah]

Evan Baker It uses the same principle

[@pocket83]

Your thumbnail makes your comment seem dissatisfied. _Skeptical Patrick is unsure..._

[@cosmicjenny4508]

+pocket83 He can't see his own forehead, again.

[@alexwang982]

THIS IS PATRICK

[@leonkan631]

Exactly Pi is this the krusty krab?

[@memawandpatrick2710]

"This trick is a masterpiece" Any of your work is a master piece!

[@Makebuildmodify]

I've always liked this one! Great explanation.

[@walkermayer2666]

I love your channel. I love how you make individual jigs for every individual process, and how beautiful the end result of your projects are.

[@JoeMarGonzales]

This type of content, I remember, is what got me into KZbin. Very informative, keep up the good work mate

[@Jellylamps]

What was that elephant?

[@starkravinglunatic]

It was PINK

[@urosjankovic1509]

he should have put it on *4:20* lol

[@GeneralSeptem]

It's in the room.

[@cyrinaefox6828]

I thought we weren't going to talk about that.

[@experiencedgamer7871]

The lamp is turned on while it's day

[@ScrapwoodCity]

Awesome as always, thanks for sharing!

[@pocket83]

Thank you!

[@deoyx]

I freaking love the little jigs you build to hold your tiny parts. the one you used to build your game made me so happy. super good work as always!

[@mversantvoort]

It's amazing how you always come up with those little awesome jigs such as the one helping you create the chamfers on the cubes perfectly!

[@user-cq8kc5es9b]

so much information you give in a very educative way,always enjoy your videos,greetings from greece !

[@nemmrrc]

Clear explanation. Very well executed.

[@JeremyCook]

Wow, that's really clever. Also, nice chamfer jig and wing nut alternative.

[@Tracks777]

Fantastic video! Keep it up!

[@pocket83]

Thanks!

[@completeandunabridged.4606]

I have seen versions of this trick many times, but it always gets me!

[@coranthianprism2442]

I have to say. I've always enjoyed your videos and this one is no different in the slightest. Stay awesome my friend

[@randyporter3491]

Watched the pinball in the cube and was tempted. Watched this and that did it ! ... Subbed ! - GREAT channel !

[@coolizmc]

thank you I have been looking for the answer to this puzzle for a while

[@dxtrs9650]

2:09 I dont know if im just really weird but that was the most comforting "its okay" ever

[@theonewithaknife1155]

THIS IS SO COOL

[@yamama259]

Awesome video yet again :)

[@giamorcos6661]

Ooo damn the motion you use is really peasant and it makes the video a lot better

[@shiny_pantz]

Thanking you for helping me understand why this is possible!!!!

[@VeteranVandal]

I remember when I was confronted with this I did the proportions in the triangle and told my friends "yeah, the problem is that this is not a triangle" and they were frustrated about that when I was a kid. "How did you know?" Well it is called similarity: en.wikipedia.org/wiki/Similarity_(geometry) I'll never forget this trick, though. It taught me to not trust vision over measurement. Also, I never thought you could make this work on wood. Guess I was wrong.

[@alexwang982]

VeteranVandal me too I was like at first the quadrilateral is pointing out and afterward the quadrilateral was pointing in and everyone was like it was a triangle

[@crimsonhorror6047]

love your videos, especially has it involves two of my favourite hobbies woodwork & Mathematics!

[@nextlevel8822]

Your small parts / complex angle / jig-creation work is incredible. I would love to see you make a set of regular-sized polyhedral dice. I may take a crack at it. That disc sander setup you had there got the wheels turning...

[@asspounderextreme69]

So glad I started following your channel. I only follow people I can learn something from.. I'm excited to see more of your videos.

[@pocket83]

Thanks. I'm glad you like. I also have a second channel, and lots of old stuff. Hope you continue to enjoy.

[@OlujaDoTokija]

Oh boy, new pocket83 video. Christmas came early this year!

[@alphamoonman]

Fascinating!

[@DiamondzFinder_]

Subscribed for the awesome work area! (:

[@AppleberrySmith]

THANK YOU SO MUCH FOR THIS

[@basedjake0261]

MIND BLOWN

[@richmond1997]

This at first looked physically impossible, but dude it blew my mind. Loved this man, cheers :-)

[@HayashiKaiji]

I like this puzzle, because it's mind blowing the first time you see it when you don't know what what to expect.

[@Dapstart]

You're like a grown up me I feel rather optimistic about becoming an adult now just graduated high school today btw :D

[@pocket83squared]

You'll do fine. Just chase your interests responsibly. Congrats ;)

[@deoyx]

Dapstart hey congratulations man!

[@mariemacfhionghuin11]

Time to face the harsh realities of adulthood: you don't get to partake in your dreams anymore, your dreams begin to partake in your responsibilities. I give you my congratulations nonetheless. Just don't screw up and you may be able to follow your dreams, unlike some of us.

[@deoyx]

Marie McKennith wow that's depressing. if you go into everything like that, things will be what you expect- crappy. but if you try to find the happiness in the small things, it will be better. its hard to do, yeah, but trust me. I've done both and the optimistic choice is always better than the pessimistic one.

[@patrice8257]

deoyx not to mention worry and anger causes wrinkles :)

[@ScottKershawRX]

The return of the Delicate Chamfer Devise! My first thought seeing the disc sander was, "but there's a better way and he's the creator of that way!" I really liked this video. Your explanation was down to earth for my 7 year old but yet stimulating to my brain. Excellent work, Pocket!

[@258thHiGuy]

I really like that the brain from the mind blown graphic is the Brain of Cthulhu and that you completely didn't intend for that to happen

[@theshinyskuntank5955]

MIND BLOWN!!!

[@RandallLakedogpix]

Love it. Thanks.

[@JstAnotherUTuber]

I finally get it! Thank you!

[@pocket83]

Great.

[@jeffcarr392]

I love how you sanded the chambers around 3 mins, I'd love to know how you figure out how to make them a consistent size, I sort of have it worked out in my head, but if you could do a video on similar but different techniques, the use of your tapered jigs etc. I love it when you explain how you did it why and the background to it. Some may think it's over the top and unnecessary, but I love it. 👌👍👏👏👏👏

[@LordFennel]

For some reason, I can't stop thinking about pink elephants...

[@258thHiGuy]

The best part of this whole video is that you have a work table that you can write on

[@pvanukoff]

Weekend project! It's bound to baffle my son. Then, to educate him. Double win. Thanks!

[@ayoutubeaccount2003]

nerd

[@want-diversecontent3887]

What did your son think?

[@haun_ted]

i love this guy

[@frunglefraggle3819]

purple elephant?

[@XGaming1]

Griffin Shea pink elephant

[@oscarmarshall3129]

Blue elephant

[@melody74u95]

Griffin Shea mayonnaise elephant

[@pattdmdj]

This completely makes sense

[@bassistbovee]

This being your most recent vidjayhoe, I have an amateur question. What is a butterfly wrench and what fastener do butterfly wrenches use?

[@alexandrumoise1511]

Also trigonometry. The ratio you mentioned is the tangent of the slope angle, intrinsic to the angle itself. The ratios being (slightly) different means the slopes of the 3 hypotenuses are (slightly) different. One of the best things about math is the interconnectivity of all the different approaches.

[@BarryIrwin]

Great video, now to go make one :-)

[@pocket83]

Do the cubes. Works way better than the flat one. Enjoy!

[@laojackos]

To tell you the truth, the amount of subscribers you have is definitely not as much as you deserve. Your content is doubtless one of the best out there and you're not as famous as other people who don't even try. I hope some day your passion and your love for what you're doing will reward you!

[@ravenlord4]

I think the oldest application of this theory was people shaving off a tiny bit from the outer edge of gold coins and keeping the dust for themselves. It added up quickly. it's possible to get a significant amount that wouldn't be detected by eye or by most scales of that day. Some kings caught on and started having their coins minted with feathered edges (that some modern coins still have today) to try to stop this.

[@neuralink7018]

Here's my guess. One of the triangles is at a steeper angle, when if you lay it out over an actual triangle (because if what I think is right this shape is a quadrilateral.) There are bumps and dips. When you switch the "triangle" then the bumps and dips combine and you get the missing square.

[@CthulhusDream]

So whats the elephant in the room?

[@pocket83]

Oh. I thought we don't bring that up.

[@CthulhusDream]

=)

[@heruhcanedean]

Cthulhus, it's that you didn't to drink your ovaltine

[@pocket83]

Fun fact time! The #1 comment on my channel is: 'dontforgettodrinkyourovaltine', followed by 'don't forget to drink your ovaltine' in #2, and 'don't forget to drinky our valtine' as the inexplicably mysterious #3. #'s 4 and 5 usually relate to not forgetting to like & subscribe, presumably from those more hasty cryptographers.

[@AdityaMehendale]

..but it's PINK!

[@aunthony2005]

Nice pink elephant in a house on a nice day... The pond looks good enough to swim in.

[@samkarnes2887]

The brain of Cthulhu in the mind blow meme 👌🏿

[@jyosiefus6773]

Nice to know i'm not the only one who noticed

[@ferus5583]

Sam Karnes I noticed it too xD

[@ferus5583]

Sam Karnes its weird

[@xninjazxmusic9757]

Thought I was the only terraria fan

[@gamingsoups]

XNinjazX Music ur not alone

[@roberthoffman9282]

Thank god, a solution!!! I thought my mind was about to break

[@elyeli6250]

What is your opinion on Keva Planks?

[@sapelesteve]

Good one! Not sure exactly how the math works out? However, it seems that when the different sized triangles are switched, they take up less space within the larger triangle thereby allowing for a slightly larger rectangle. I gotta make one of those..........

[@Hextator]

they take up less space because they took up less space to begin with the space just gets moved from being a long rectangle to the (top) sides of the pieces to being a square in between them

[@McHeisenburger]

The new Minecraft update looks great.

[@Me-sv2xe]

Name not found You are damn right !!!!

[@leonardochtz3734]

Nice!

[@phaedruscj3330]

The cube part is easy :) especially if you buy square dowels and forego the sanding jigs and just hold the cube at an angle to the sandpaper. But How do you cut them at the angle? The table saw sled cuts only right angles correct?

[@shrknan0145]

Wouldn’t the area for a triangle be a=1/2(bh) because if you do a=1/2bh then you would have 1/2b•h which is not equal to bh/2

[@lewismegablastar18]

my dad had pictures of that puzzle every were and he always talked about how it was amazing

[@Duraltia]

I'd like to see this to be performed with precision machined pieces made of hardened steel without any visible gaps at all times. Good luck =)

[@user-rg7ru3lo9w]

the sharpie on that table doesn't look like it will go away

[@yssa2345]

Cool video. BTW what is the make and model of your tablesaw?

[@pocket83]

It's a RIDGID something or other. There's a video or two about its specifics on my second channel.

[@yssa2345]

pocket83 Thanks! I'll check them out.

[@cosmicjenny4508]

Those looping GIFs of the 'infinite chocolate' trick have a cheat to them. As the parts move, they sneakily grow to continue the loop. ;)

[@moonlightbuildscy4325]

Great video! I'm interested in making it myself. Did you just use a crosscut sled to cut the joined cubes into triangles?

[@pocket83]

I used a bandsaw. I was conservative with cutting near my line, and then I disk-sanded- it in order to sneak up on it until it was a perfect fit. Sorry that I left that bit out. I wouldn't recommend using a sled unless your sled has toggle clamps to hold the piece, or unless you make something for the piece to sit in first. Don't rule out using either a coping saw or a backsaw. The pieces aren't very big, so being careful is better in this case. Good luck ;)

[@moonlightbuildscy4325]

Thanks for the reply! Without access to a bandsaw I'll likely use a coping saw like you recommended. This will make a great gift for a math teacher once I graduate.

[@lepoloo]

What was the purpose of the pink elephant? I precise that I'm french and I'm not sure I get everything that's said all the time. :) I knew this pattern and it's on my FB page since several years.

[@AlZimmermann]

I have a question about the shape of the interior of the tray. Do you think the hypotenuse should be a straight line or should it be slightly bent to be able to fit the larger configuration of tiles more snugly? I can see advantages to each.

[@pocket83squared]

You might be right. I honestly don't remember if I did something like that, but it sounds reasonable to make the tray kind of in-between those two shapes. It might make a more convincing illusion, since the play would be halved in both cases. I do remember that this project takes some fitting. And I also remember that the cubic version was more convincing.

[@tonyhoang987]

I would have just gone with the law of conservation of mass as matter cannot be created nor destroyed, but I like the visual proof of physical squares.

[@thankyoujesus1143]

I noticed the small crease at the edge but still nice thing it is way better then the candy one

[@notfuz9218]

Gonna 3D print this ;)

[@philiptripikelis2395]

Great video as always! May I ask, what type of wood is the one you're using? I have some of it myself and I don't know how's it called.

[@pocket83]

Thank you. The cubes are made from yellow pine.

[@philiptripikelis2395]

pocket83 Thanks!

[@thecreator6959]

When he told use what the missing cube was my mind was blown

[@3mar00ss6]

what's with the purpel elephent in the black rome

[@NonTwinBrothers]

I'm gonnaaaaa watch all the videos on this channel, newest to oldest, HAHAHAhaha

[@cptlorkel1556]

i learned nothing but it's surprisingly claming

[@pocket83]

For as smart as you must be, you should easily learn to spell better.

[@cptlorkel1556]

I meant i was just kind of listening to it while i was doing something else, and wasn't really paying attention, not that i'm smart.

[@pocket83]

I'm just razzing you. People often tell me that I have a soothing voice. Ladies think it's _smooth._ ...I mean, I'm sure they probably do.

[@cptlorkel1556]

I can confirm this as i am actually in fact a lady.

[@dontreadmyprofilepicture931]

It's simple, the smaller triangle makes more space left to right making an extra square

[@Skullmonkeys44]

person1-how that even possible ? person2-so, you see there is [..] person3-SHUT UP, IT'S MAGIC !!!

[@oriondaniels4760]

I really do love your content! I still can't tell if you are 25 or 55, in any case keep up the good work! :)

[@pocket83]

Thanks. I'm in between those somewhere ;)

[@NonTwinBrothers]

those things where so easy to figure out

[@DeadlyPants1404]

I really want cubes now

[@aquaman6166]

The moment he said "let's think of it as the fraction 3/4" I was lost

[@andrewandrews8073]

Writing in sharpie on your table? Pocket83, you madman!

[@pocket83]

Wipes off with denatured alcohol. I keep it in a spray-bottle.

[@bno112300]

Also wipes off with more sharpie. A great trick to know when you grab the wrong marker at a whiteboard.

[@XxSteamStreamxX]

8:48, you used the Brain of Cthulhu from Terraria :O, respect.

[@dersjuanpablo7268]

At 9:38 how did you get an overall ratio of 5:13? It looks as if you tried to add the fractions but didn't know how to

[@pocket83]

Pause it at 9:49. You just count the blocks in the overall puzzle. It has nothing to do with adding fractions.

[@vic5254]

If this type of math interests you I highly suggest the Vsauce video on the Banach-Tarski paradox. very interesting stuff on a similar topic

[@yumlbena]

this helped me in my maths test

[@stevenmcculloch5727]

Ever think about getting into 3d printing?

[@M4TTM4N10]

It kinda just seems weird to me, the idea that a woodworker would get into 3D printing. Why learn all this woodworking stuff just to then print stuff (unless of course printing is the only option, but I can't think of anything off the top of my head that would fit that category)

[@sevferaly]

Haha brain of cthullu at 8:50😂

[@JayFolipurba]

do I see Prime divided by Fibonacci? 2, 3, 5 by 5, 8, 13

[@tommybomby4122]

wait but i don't get it, in one arrangement it fills the entire puzzle thing but in another arrangement it nearly fills it up that's impossible, you would have to compress one of the peices

[@__beren__erol8942]

Wow

[@Cloakarius]

8:50 What a Brain of Cthulhu

[@cameronwilliams7667]

im not sure what you just explained... how does that solve the puzzle?

[@pocket83]

The three different ratio numbers mean that all three triangles are different shapes.

[@cameronwilliams7667]

pocket83 i understand that, but how does that help solve the puzzle?

[@pocket83]

This is a case of Poe's law. Are you really having trouble with finding the solution to a four-piece puzzle? Even puzzles with _SpongeBob_ or _Dora the Explorer_ on them usually have nine pieces.

[@asdfasdf-tp7fm]

watch the end of the video.

[@AshLeeeeee]

the triangles are vary very slightly