An Amusing Paradox Most People Cannot Figure Out - How Did 1 Line Vanish?
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- Опубликовано: 13 сен 2024
- This is an easy trick but most people have a hard time explaining what is going on. In this video I will turn 13 lines into 12 lines, thereby "proving" 13 = 12. Where does the extra line go?
My blog post for this video:
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Source
Baumback, Randall R. Mathematical puzzles for the secondary mathematics teacher: a collection, classification, and evaluation. Diss. 1980. Puzzle 69. csueastbay-dspa...
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I'm proud this channel probably has the smartest subscribers on RUclips. Let's be positive and encourage math education. This puzzle comes from a dissertation in 1980 by Randall R. Baumback for a Masters of Science in Education. This puzzle comes with the quote: "Here is an amusing paradox that most people find hard to explain." The introduction explains: "An evaluation of the puzzles was made by five secondary mathematics teachers...revealing that nearly all of the puzzles were rated very highly. Hence, the puzzle collection was judged to be an excellent resource for secondary mathematics teachers..."
Source:
csueastbay-dspace.calstate.edu/bitstream/handle/10211.3/7572/Randall.BaumbackThesis.pdf
Kasîm Dude, you can't even spell 'multiplied'.
Scott Moore The sarcasm went right through you, didn't it? Proves his point.
Sadly it seems to be human nature to start bragging and mocking others as soon as you solved a puzzle, be it easy or not. Grow up. Everybody can look at the wrong detail and some are more interested in taking a walk in the forest.
Sir someone said this to me in quarantine :-
0 = 0
27 - 27 = 30 - 30
(9 × 3) - (9 × 3) == (10 × 3) - (10 × 3)
So by distributive property
9 × (3 - 3) = 10 × (3 - 3)
We will divide (3 - 3) from both sides
So, 9 = 10 × (3 - 3) / (3 - 3)
The (3 - 3) terms will be cancelled so we will get 9 = 10
Is this correct Sir ??
Someone has sent me this and said : "I proved 9 = 10"
@@IS-py3dk ik im late but when you divide (3-3) from both sides you are dividing by zero which is illegal
Fill upp 10 small water buckets. Pour all the water into one big water bucket 10=1
😂😂😂😂😂😂😂
🤣🤣
😂🤣👌👌👍👍
Me sitting here in quarantine looking at such funny comments makes me laugh cry and have a red face 😂😂😂😂😂😂😂😂😂
It would be more like, distribute one water bucket among the other 9, and bam! 9=10
Here are two lines
- -
now I push them together
--
OMG IT HAS BECOME A SINGLE LINE. I just proved 1=2. Amazing.
Treufuß here is a line, now throw the paper away, 1＝0
You killed me. LOL !!!!
Illஇsion
Funny ;)
Hating Mirror Looooooooooool
How tO pRovE tHat 1=2. TakE A liNe aNd ThEn cuT it iN haLF, Now yOu haVe 2 LineS, sO 1 mUSt eQuaL to 2.
make this top comment
+1
Literally.
This was my thought process
Whahahhahaha this comment is gold
O
As he has said before, the content of his videos varies from very easy to very hard. He wants the videos to be hard enough to have a challenge but easy enough for us viewers not to be stumped. Well, he's human, so he will always have a time when he makes it too easy or too hard. Let's not be too rash about this puzzle. I quite like it :D
illuuuuuuuuuuuuuuuuuusion
Scott Moore RUclips algorithm. I dont know if its still like it used to be, but some years ago the more comments you made, the higher you were in the comment section.
Scott More I guess your likes disappeared like the 13th line.
Do you know how often I have found the same things about my comments? I write something and got no likes, later someones comes with the same comment and gets 200 likes.
Guns N Roses doesn't even have 13 albums, not even 12, not even near.
Give him a break. Most of his puzzles are hard. What's wrong with a quick fun one? It's not a big deal.
Exactly, stop hating, at least this is interesting
No. He should know better than to teach. He must suffer.
I like this problem very much. It's a great introduction to the visual biases of our brains. I would definitely show this to my friends as a conversation piece/starter.
I wish I had people that I could geek out with about maths. :(
+ Nigel Wilcox I share your attitude. One conversation or line of enquiry started by this problem for me is if you slide the upper piece downwards in the opposite direction you get an extra line "mysteriously" appearing out of nowhere, though not all of the lines are correspondingly shorter.
Other Mind Your Own Decisions vids illustrating visual biases include the circle illusion, and the missing square illusion. Again, what's interesting isn't just the particular question and its answer, but the range of similar effects. The circle illusion leads on to the pendulum wave effect for example, since both consist of complex flowing patterns formed by simple individual back and forth movements. The missing square and vanishing line puzzles both exploit our tendency to ignore slight differences of measurement under certain conditions.
(I also think the vanishing line effect in this vid is interesting enough without describing it as "13 = 12")
Step one: Click the little gear at the bottom of the video and set the playback speed to 0.5
Step two: click this - 1:46
Step three: ???
Step four: Profit
oh my...
Set it to 0.25
Creepy...
Superrrruuuuuuuuuuuuuu
THE ILLUUUUUUUUUUUSION SOUNDS THE SAME LENGHT EVEN THO ITS LONGER
JUST LIKE THE LINES
Answering at 1:10
This is a famous old puzzle. It was once known as the Vanishing Leprechaun trick, also known as the Disappearing Dwarf trick. Those ones ornamented the basic trick with lavish illustrations, and apparently fooled many people. Newspapers have run essay contests asking people to explain how they think the trick worked, and judging by the poor insight of the winning entries, few people got it.
The basis of the trick lies in the phrase "Where did the missing line go?". Once you are tricked into thinking in terms of entire lines shared between the papers, it can seem mysterious. A little, anyways; it was probably more challenging with illustrated leprechauns.
But of course there are no physical lines shared across the papers. Here we are always taking a "line" to mean one of those all-the-same-height things that we can see when the papers are put together and not the more technical meaning of "line".
Physically, there are only partial lines of differing length. We have two ways of joining them into "lines" of all the same height:
Method A: join the shortest to the longest, second shortest to second longest and so forth. For N partial lines on each paper, this gives N lines. In this case, 12.
Method B: Don't join the longest to anything, join the second longest to the first shortest, and so forth. For N partials, this gives N+1 lines because the longest ones don't need a partner. In this case, 13.
This basically explains the trick; all that's needed is to realize it geometrically: Space the partial lines evenly, make the cut straight, etc.
Am I the only one who thinks that he is quite more passionate and funny at this video?...
... or it's just another ILLUUUUUUSION???
This comment is SOOOOO underrated
This is very similar to the infinite chocolate bar "Illusion"
I thought this was a serious maths channel, but that was just an illuuuuuuuuuusion.
This one is trivial. It's just avoiding dividing the two outermost lines, which is essential. It's just created 12 slightly longer lines (one and one-twelfth times the original length).
After this video you'll have the Vanishing Subscribers Paradox 👻
It's an illuuuuuuusssion!
1:45
0:43 I see it already, the lines on the edges are at only one paper and they combine to lines that are a little longer
12 slightly longer lines = 13 slightly shorter lines
but 12 does not equal 13
But just think of how many lines are in the illuuuuuuusion. That's a lot of U's with extra lines.
It can be explained in another way. When you draw the diagonal, it starts from the tip of the first line leaving no excess. Similarly, it ends at the last line on its tip. Which means the protruding pieces on the other side of the diagonal are only 12 . So, when you move the pieces, it just goes and adjusts against the first 12 lines. The last line appears to have vanished as there was nothing protruding there.
And, as he said, the length of that line appears to have distributed for the other 12 !!
I know it's tough to come up with problems to solve, but this one is really dumb.
Much, way much more interesting is the Lewis Caroll geometrical paradox (involves cutting a rectangle into different shapes and matching them to form a square with a larger area)
This is probably your worst video. And that's hard for me to say, because most of your videos are really good and quite interesting.
friedmanism Surely that makes it easier to say?
I agree. Wouldn't that make it easier to say? When you said it would be hard to say, I assumed you meant all his videos are bad.
I don't think you understand your own words.
Its fucking crazy. Why the fuck would someone ask this question. Im going through the comments puzzled myself. It made me question why i was subbed in the first place, but he had some other vids that were good.
Well, it'd be harder to say because you don't want to be discouraging and destructive, but also easier because "most of [his] videos are really good and quite interesting" and this one doesn't even come close to that (more accurately, easier to _decide_ that this is his worst video).
he do this kind of stupid thing every once in a while
A little kid could see how this was done in seconds. Can we have actual problems?
We don't. The problem is a tedious mathematician having to agree with our explanations.
Minecraftster148790 He has very few mid-range puzzles. They're all either absurdly difficult or absurdly simple. But this takes the biscuit: so simple it doesn't really qualify as a puzzle at all.
I actually believe you don't get the point of this channel at all. I feel like what really makes this channel stand out from the others is exatcly its universal appeal, you can learn beautiful mathematics of all levels here. All he's trying to do is to develop a sense of mathematical logic on the viewers, maybe you think this is way too easy for you, but you never know when a hard problem that can be easily solved by a bijection involving this, for examlple, will appear. If all you care about is evaluating integrals or mindlessly applying theorems you should be reading a book, not searching for it on youtube.
Did you see my comment? If you can solve it, I will eat the phone I am using to type this comment.
wimpykidfan37 no, could you just copy paste it into this thread
Draw 13 lines next to each other.
Move the last line on top of the first line.
You now have 12 lines.
...
13 = 12
You could repeat that for smaller/bigger number and it makes that any numbers is equal to any other number
Its the same amount of lines. The 12 lines are a bit bigger and therefore the extra bit adds up to 13
If you are saying the sum of lengths is equal in both situations, I would challenge you to prove that!
@@oscargr_ Lemma: If you cut 13 equally distant straight lines, numbered from 0 to 12 from left to right, of the same length with a single straight line G, starting at the top 0 and ending at the bottom of 12, then the line i is split into 2 lines each with a length of i / 12 * L and (12 - i)/12 * L, where L is the length of each line.
Proof: Using an appropriate coordinate system we can assume L = 12 and the distance between adjacent lines is 1. Then the intersection of line 0 and G is (0, 12) and the intersection of line 12 and G is (12,0). As G is a straight line it is therefore described by y = -x + 12. Line i consists of all points of the form (i , b), where 0
I love how people in the comments are always complaining that the math problem was either way too easy or way too hard.
Ok paused at 0:57 and went to sleep. Not filling my head with this nonsense right before bed. I came lookin for integrals and stuff and got the reason why so many US kids are failing school.
Then you slept for like less than 10 mins
me 2
it is not nonsense. it is useful in its own way.
not useful bro. this is pointless and not a puzzle. definitely not what i expect from this channel
I expected this and much more from this channel
I would say there is one less line because the lines are longer and that extra length from all the lines equals to that 1 missing line. It is like the infinite choc bar trick.
*I L L U S I O N*
Here is a puzzle that you might remember from the same book that you got the Secret Word puzzle:
There are three brothers: Al, Ben, and Carl. Two of the three brothers are boxers. The shorter of Al and Ben is the older boxer. The younger of Ben and Carl is the shorter boxer. The taller of Al and Carl is the younger boxer. Which brother is _not_ a boxer?
Ben is the only non-boxer. Al is the older boxer, Carl is the shorter boxer. All of the other possible combinations lead to logical inconsistencies.
+NSNick So does that one. You have Carl, the shorter boxer taller than Al, the other boxer.
Al is the younger, taller boxer, Ben is the older, shorter boxer. Carl is a dachshund.
Steve's Mathy Stuff My exact conclusion. If you proceed under the assumption that Carl is the younger and taller boxer, both sequences of either Ben or Carl being the shorter boxer reach illogical conclusions. Therefore, Al must be the younger and taller boxer and the rest plays itself out.
One of the easier ones you posted. Or, it could just be that after watching your channel so much, I became that much smarter. LOL! Please keep these going. You inspire people to think about math. That's really needed these days.
This guy gets so much hate, it's beyond ridiculous.
Just because he makes some videos that aren't up to your *prestigious* level, the videos aren't acceptable? Maybe there are people in this world who want to try these types of riddles for themselves, but aren't up to the challenge of the harder mathematical problems he uploads. Sure, I thought that the solution to this problem was simple, but for some others, it may be a good introductory puzzle to bring them into the world of logic and mathematical riddles. I can't stand commenters who hate on virtually every riddle video like this (even the extremely well-made TED videos) because it's not perfectly suited to *their* individual needs.
Seriously, take your egos somewhere else. This is a math channel, not a competitive forum to see who can be the most insensitive of others.
Each line gets a little longer (prob. by say 1/12th of a line). The total CUMULATIVE LENGTH of the new 12 lines is equal to the total CUMULATIVE LENGTH of the original 13 lines. The quantity of lines is irrelevant, it is the total length which matters. In fact you could get the new shape and re-do the same process recursively until you are left with, in the limit, 1 single very long line which has a length equal to the cumulative length of the original 13 lines.
I've seem this before with dwarves, it's way harder to see what's going on then.
just that as you take the top half to the left the line gets overlapped with its immediate left.
also that the tip of the first line touches the diagonal 'cut line' , it doesn't leave any trace/part of a broken line if you take it to the left .
so, because of this overlapping of each line segment ( except the first which is completely in the fixed,bottom half), to its immediate left line segment, the number appears to get reduced by one.
So many great videos, and then all of a sudden a video about a trick you'd do at the bar to win a bet against drunken idiots. Dammit, Presh, you're better than that.
Why does he keep doing this type of video despite all the dislike? RUclips for dummies?
"math for dummies"
Thales F probably because not everyone is actually smart enough to understand this.
The same kind of problem was relatively popular but the other way around and instead of lines it was about chocolate. It was about cutting and rearranging chocolate so you get more than before. It's only really working if you have objects with a higher quantity otherwise you see the difference. It was also done with pizza where you get one extra slice without anyone noticing it.
It can be useful to get an advantage at dividing food but only if no one sees you doing it. The illusion is only that we can't compare before and after.
hes allowed to make the videos he wants despite the fact that you guys dont like it
From the thumbnail: Nothing "vanished." 13 lines of length L were cut and rearranged into 12 lines of length 13L/12.
If you want to ask where one of the lines "went," then the original rightmost line became 12/13 of the new rightmost line.
You can see this better by starting with just 2 lines. Draw a cut from the top of the left one to the bottom of the right one, and slide the part above the cut up-left until the lines align.
Now there's just one line where there were two, but it's twice as long.
With 13 → 12 lines, the length change is much more subtle, so it's easy to miss.
Fred
The missing line is an ILLUUUUUHHHSIOHN
I followed along with you, but it did not happen.
Because the diagonal didn't go above the 1st and below the 13th line.
The length of the 13th line got distributed among the 12 lines. The animation does not show this but when you slide the paper up, the uniform length of each of the 12 lines will not be the same as the uniform length of each of the original 13 lines.
So if we have, for example, *1001* lines and we do the same trick, the illusion of the missing line will become even stronger as each line will only be longer by 1 / 1000 of its original length.
We can generalize this: if we arrange _n_ + 1 lines in such way, the _n_ resulting lines will increase by 1 / _n_ of their original length.
In other words:
x+1=x?
1/x= increased length
The one line just "diappears" in the length of the remaining 12 lines. They all will incease by the same amount. Easy. You can do that again and again, untill there will only be one very long line left. And that line will have the added length of all the 13 original lines. Nothing really vanishes. It just "transforms".
I like the change of pace in this video.
using reductio ad absurdum, try this with two lines in the rectangle which results in a single line that is clearly much longer than the two original ones.
So glad I have started watching this channel. I am getting smarter by the minute. Naturally I had seen this before, but it was somewhat fresh to me. I can't believe that some people can't figure it out in seconds.
Skip the video from 0:34 to 0:44 by double clicking on the right side of the screen for tablet /phone / ipad, and skip the video from 0:34 to 0:39 by clicking the right arrow in the keyboard if your in computer. Notice how the lines lengths have changed.
I am glad I figured it out that they were longer. That was the first thought that appeared in my head. Guess I am lucky.
It is because each line has been collapsed to a slightly longer line to the left.
When the comment section is full of people that understand it why does the channel get hate?
Now, let's cut all thirteen lines up and rearrange them on top of each other. And, remarkably, now twelve of the thirteen lines have disappeared, leaving only one line. *MINDBLOWN*
Before I watch the rest, the total length of the lines is distributed over 12 lines rather than 13, so it makes 12 slightly longer lines
Nope, the first line is completely on the bottom paper, and the last line is completely on the top paper. When you shift the paper up and to the left, the top of the second line goes on top of the first line, making the first line longer. For the last line, we can see that if you were to turn the paper upside down, this would appear identical when cutting, and the last line (on the left now) grows from the second-to-last line. There you go, the 13th line breaks apart to make the other lines longer.
As my teacher used to say "It should be readily apparent to even a casual observer!"
Adjust playback speed at 0.25 and click on 1:46
Enjoy
Well, at least I got this one even before he actually slid the lines.
The infinite chocolate paradox is on the same idea but in a slightly different way. The chocolate's pieces will always add up after cutting and moving the chocolate in a specific manner. The catch is that the size of the piece is getting smaller and smaller. The change that happens in 1 step is not detectable easily.
That’s what I thought of!
1:46 ILLUUUUUUUUUSIONNN
Huh? All the lines are longer now, how is this fooling anybody?
'I immediately figured it out. Now how do I put this in English?'
Here in comment section people are making fun uploader but still great efforts and keep it up
I really like this channel, most puzzles and riddles are very interesting. But to say that this will fool most people is just wrong, i dont know anyone that would be fooled by this
The line on the left doesn’t split into two or move and the one on the left doesn’t split into two and becomes part of a new line, removing one line. If the line on the left was split into two, then it would add a new line, cancelling out with the other line.
I now have 2 in a row that I could solve. I feel like a lotto winner. Actually this was rather simple, but a least I got it in seconds.
Yea. This was so easy I thought there was a trick, so I stayed for the next 3 minutes making sure that I didn't miss anything lol
If you form the question like: how can you make 12 lines starting with 13 with a single cut, it's probably better, and many will likely fail.
If you have some tape,you can keep rearranging into the shape until there's only one really long line
into making the 12 lines longer . Still there , just beside itselves .
incidentally , the lines are 1/11 longer (not 1/12)
13 x 10" = 130 ,,, 12 x (10 + (10/11)" = 12 x 10.909" = ~ 130
best video in Internet ever, great work keep it going!!! like, subscribe and share!!
The reverse of this process (i.e. 12 into 13) is how people used to counterfeit money (paper notes). You end up with an extra note, all with slightly shorter width. This was why serial numbers were introduced at the corners to prevent this method of counterfeiting.
Why doesn't anyone mention that you are simply covering up one line.
The two lines at the ends can not be seen below or above the diagonal.
If you did not shift the paper along the diagonal but along the long axis of the paper, you would also have 12 lines.
Would you then ask where did the 13th line go?
You can make this solution more interesting by proving the sum of lengths of the 12 lines is equal to the sum of lengths of the 13 lines. (For all possible lengths of the lines and distances between the lines)
there is no missing line. the twelve lines after making the cut are all a bit longer, and their total length is the same as that of the original thirteen lines.
Did not everyone figure this out before the question was asked? It was clearly seen during the slide that short piece of the second line is sliding toward full first line
The one missing line is "being added" to the other 12. They are 1/13 of a line longer
This illusion is often encountered with the lines subsituted by something some suitable figure / picture / representation like leprechauns, dwarves... The most effective way to solve those is like in the video : add up the heights of all the figures from 'before' and 'after', pay a close look at details on specific figures and voilà.
Thanks for this truck...Now I will amaze my cousins with this magic...
This is just like that old “magic trick” that summons a new piece of chocolate by cutting a block!😂😂😂
Love how many people were fooled by it, including me until I detected the trick🔥
For our next trick: we're going to take 156 candies, grouped in 13 equal piles, and then MAGICALLY move them into 12 equal piles, but still have the same total number of candies. MAGIC!
wtf, isnt the shpae just diferent, and if the shape is diferent no 13=/=12
Bro you cant say that 13=12. From the rectangular piece above, the diagonal cut is at the end of the 1st line and the 13th line. When you move it, you can clearly see that a bit of line is added to the 1st line although the length is only up to the diagonal cut line. Let's put it in an algebraic expression => 13x=100%. Then, the new one will be a bit longer and the 13th line is distributed so => 12(x+x/12)=100%. You can't say that 13=12 because of the unequal length from before and after. By the way, I haven't watch the explanation. Thank you :)
The 12 lines are taller than the original 13 lines. If you measured one of the 13 lines and compared it with each of the 12 lines you'd find the total difference of height each of the 12 lines has compared to the original 13 lines would add up to the missing line. I'm saying this because when you cut the diagonal there were very clearly extra pieces of line in the middle of the paper with the ends being the only pieces completely whole, when you move the diagonal into place the extra bit just goes on top of the whole line, thus making the line taller than the original 13 lines.
it might have been cool if the explanation shows the sliding part with each line in different colours :)
Anyone notice that the lines become longer? What you did was because at the more acute base it was cut below the line
I believe it is because the libes get a little bit longer, as it would do with a rectangle with 2 lines
LOL I figured this out right away, even while drunk. The trick is that you cut the paper in such a way that the two end lines are not cut at all. They are complete lines. Then you slide the partial lines together, and end up making lines that are actually longer than the original lines. Very clever illusion, but it is of course A LIE. LOL
My answer was that when the shape is cut diagonally, each half has 12 lines.
The 13th line is divided equally into the other lines, so the line is still there, just within the others.
I think i got it before the solution . If I’m wrong, don’t hate me, are the lines longer? 1:08
it's simple! you don't cut first one and last one. and when when you slide them the rectangle is be wider than older
The 13th line was put on top of the 12th line, and the lines in the second shapeare slightly taller.
Cut the paper in half horizontally and place the two pieces side-by-side and you now have 26 lines. OMG 13=26.
It gets contributed to the other lines. I'm sure if you measured the distance there should be a discrepancy between the one with 13 lines and the one with 12 lines. Those small discrepancies accumulate.
I played the part where the we slide the upper part and it appears as it thirteenth line coincides with the 12 th one but the 12 th one part coincides with the 11 that and similarly the second line part adds up to the first line so the first line is added up in its length.
im writing before the answer to try
the first line is not really cutted in length and you slide the tip of line over it
same way the last line is "fully" cutted and slided on top of tip of line 12
so basicly 12 longer line...
this is similar to if you had an 8x8 square and when rearranged into a 13x5 rectangle, so 64 (8x8) = 65 (13x5)? Can you please explain this in another video if you haven't already.
You are line to us.
Technically there are 11 extra lines by cutting the paper, also separating one full line. These 11 have been joined to the left most lines on the other half plus the whole line on the upper sheet has been added to the 12th line on the lower.
Fuckin easy.
The disappeared line made all the remain lines look taller. That’s it
is anyone surprised that the flat earth crowd never visits this channel?
"This problem fools many people" - 'People' who can't figure this out should not qualify as people in my eyes... The animation at 0:36 literally *shows* the lines becoming longer...
spoiler alert
That one line didn't vanish. It just had it's length divided among the other lines.
So it's not 12 = 13, it's 12a = 13b.
what software is used in creating this video?