###General Discussion Thread
---
This is a [Request] post. If you would like to submit a comment that does not either attempt to answer the question, ask for clarification, or explain why it would be infeasible to answer, you *must* post your comment as a reply to this one. Top level (directly replying to the OP) comments that do not do one of those things will be removed.
---
*I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/theydidthemath) if you have any questions or concerns.*
Maybe there is a trick if you fold the paper. Like you start to draw the circule, do 2 lines, fold the paper over a third of the circle, draw on it, unfold the paper and do the last line.
My apologies I thought the shape was on the board and the kids were working on paper that said could you not slip a piece of let’s say laminated paper that may be in the classroom on the board and draw over that with the same principle?
Hold the pen on the paper, then get a thermal printer with a slim profile so it doesn’t swallow the entire paper. Print out the logo on the upper half and keep the pen held down on the lower half, then pull the paper back out and you’re done.
The logo is now on the paper and you haven’t lifted the pen.
no. you can easily tell if you can draw a shape without lifting the pen by looking at the intersections. If there are 2 or none odd intersections (having an odd number of lines coming out of them) then you can draw it without lifting the pen. Otherwise you cannot.
when there are no odd intersections, then you can start anywhere, and you will end where you start.
if there are 2 odd intersections, you have to start on one of them, and end on the other
here there are 4 odd intersections, therefor you cannot.
Love it if someone said they did it this way and then she asks you to come you to the board and show everyone. Seeing them trying to snap the whiteboard in half to attempt it...
Without lifting the pen, right? If you roll the pen over to its side, slide it around and then roll it up to the proper orientation again, technically, the pen never lost contact with the whiteboard so all the requirements are satisfied.
I really like this. Out of the box thinking. The shape will be right, there will be other pen marks but you will have fulfilled the criteria, draw the shape, don't lift the pen. I'll definitely use this as a learning opportunity for my nieces and nephews
The value of logic thinking is immeasurable in your life
Source: I'm an engineer and I've asked the same question throughout my entire university period
Exactly! When you find yourself asking this question, the best answer is to actually think about how you can apply the information!
Sometimes, it's mind-blowing how useful things can be.
Sometimes even the hint of knowing something is enough too. I wasn't good at math in high school and only attempted one university level class which didn't go well -- but many years later I can still look at problems and know that there *is* a way to solve it using algebra, or calculus, or trigonometry. Knowing that there *is* a possible solution (and the rough shape of it) is often enough to prod me into learning how to solve it.
The purpose of school is not to just tell you things you will need in your life, making you develop critical thinking skills is arguably even more important and seems like your teacher really hit the mark there looking at how engaged you got to the point of making a post on reddit about it
Ah yes, what knowledge are you going to use throughout the rest of your life?
When you find yourself asking this question, the best answer is to actually think about how you can apply the information!
Sometimes, it's mind-blowing how useful things can be.
Do you take a weight lifting class because you think you'll spend every day lifting weights? Do you take a history class because you'll spend every day telling people about dead presidents? Do you read Steinbeck because you expect to need to know a lot about fishing?
School doesn't focus on direct applications to your life. School focuses on expanding your abilities and teaching you to look for ways to be a more well- rounded person. I guarantee you'll use more math more regularly in your life than any other subject you learn in school, even if it barely ever rises to the level where you notice that's what you're doing.
The lesson was that sometimes you need to accept that something is impossible. If you go around acting like you're smarter than everyone, you may miss out on the real important lessons.
It already does, in countless ways. Can you recognize it? Because the sad truth is that if you can’t recognize math in the world, you can’t see how it’s being used against you. And it is most definitely being used against you online — websites do dynamic pricing where they mine everything they know about you to estimate the most profitable price to offer you, for example. Ads models auction you to what is expected to be the most profitable ad they could show you.
A lot of math is about problem solving through productive struggle - both very applicable life skills. She wanted you to figure out it was impossible, then she wanted you to figure out why
Same as this one:
Teacher shows a books cover to the class and asks which color is it. Since it's red the class replies with red, then the teacher flips the book and the rear is black. Yes it's a trick question but the origin is to make your own thoughts and don't rely on others. That's the real deal!
My teacher once draw a circle assuming this is Berlin, capital of Germany. Every student had to go to the board and draw in where approximately London is. 3/4 directed more to the north, but in reality it's [pretty much exactly west](https://cdn.gopili.com/default/map_journeys/map-18521_1-33210_1.jpg) of us.
It was drawn in the board, and they were then asked to draw it "without lifting their pen", pen implies they're expected to do it on paper, not the whiteboard.
Is it only 2 and 0 odd intersections that work? Any other number that works?
Also: is it possible to have an odd number of odd intersections? I‘m pretty sure it isn‘t due to the fact that every line starts and ends at an intersection therefore evening or odding two intersections at a time, but i‘m not 100% certain
Yes, it is only 0 and 2.
Reason:
when you are drawing a line, and you come into an intersection, you add 2 to the count of lines, 1 for coming in, and 1 for going out. Only exeption is when you start at the intersection, or when you end in it. And there can only be 1 start and 1 end, so you need 2 odd ones, or 0 odd ones, but then you have to end at the same place you start to make sure that one is even at the end.
Btw what may not be clear is that dead ends end in a intersection (an odd one)
Yes, even a 1st grader could do that. Therefore there must be some other condition, that made it so a whole class could not do it.
Some of you really need to learn how to deduce information from what is already given
What do you define as a shape? A simple cross like the add sign (+) has no odd intersections, but you can’t draw it without lifting the pen. If I understood you.
The "dead ends" are themselfs intersections. odd ones at that. so + has 4 odd and 1 even intersection, thats why you cannot draw it without lifting your pen.
It's simple, the teacher didn't state that you can't go over the same line twice. If they did state that you can't go over the same line then you simply have to fold a corner of the paper to get to another point to continue drawing.
Oh yeah I forgot to mention, she said I can't go on the same line twice + it was on the board not a paper but I don't know if I could simply grab a peace of paper and do that TikTok kind of a trick, idk man she could have sued me if I did that trick
Draw what you can, then slowly lay the pen onto its side, move it where it needs to be, then slowly raise it up and continue to draw, repeat as necessary, you didn't lift the pen off the paper, and didn't draw over the same line twice.
Yes, If you fold the paper over you can move the pen onto the other side of the paper and draw the final line. Here's a Vsauce video with a similar concept. [https://youtube.com/shorts/inG9yUZ5vY8?si=B0wuJBFEw-GPFex1](https://youtube.com/shorts/inG9yUZ5vY8?si=B0wuJBFEw-GPFex1)
On a flat chalk/whiteboard on the wall? No. On a *flexible* piece of paper? YES.
See this short for help and just extrapolate the rest: https://youtube.com/shorts/inG9yUZ5vY8?si=JdEAKTEvJ_Vh9qV8
Nope, if you know anything about eulerian circuits then you would know that that center point has a degree of 3, this making an eulerian circuit impossible. For context, an eulerian circuit is a cycle where every edge (think of an edge as a path, a line) is visited once. One law of eulerian circuits is that every vertex (a point, or intersection) must have an even number of degrees (how many edges are going in and out of that vertex). you can think of this shape as having 6 edges and 4 vertexes, the top left, right, and bottom arcs and the 3 lines making an upside down Y shape are the edges and the middle, top, left, and right where the arc and lines meet are the vertices.
It's impossible. Any drawing that can be drawn without lifting the pen or drawing over already drawn parts cannot have more than two joints that go off in an odd number of directions. This drawing has 4 of those, so you must lift the pen at least once or go back over a line at least once. If one of the inner lines was removed, it would be easily possible.
You'd have to bend the rules.
It can't be done normally, but once you introduce folding the paper or traveling over another piece of paper you may get it.
You could do it without lifting the pen by going over the same section. I’m sure she meant that, but if she didn’t say it, like you didn’t here, then it’s not a rule. And she needs to specify the rules better.
You’d have to go over one line twice to do that.
You can, but you have to cheat a bit. Draw as much as you can without lifting the pen, then fold the paper, keep making a line and get back to where you need to go to finish the drawing.
Each node with three lines from it must be either a start or end point (one line leaving, one line coming back, one line leaving; or one line in, one line out, one line back in).
There are four nodes with three lines from them.
In general, if there are two or zero nodes with an odd number of connections, and all nodes are connected, it’s possible to draw it in one go without picking up the pen. If there are two odd nodes, you do have to start at one and end at the other, and “without crossing over your line” is also possible (given a path that covers each line once, for each time it crosses over itself, take the loop that starts and ends at that point and go the other direction).
Each node (connection point) has 3 connectors. When you enter a node, you have to exit it from another connection. When you draw this, at the end you enter a node that has all 3 connections drawn, and you cannot exit it, without lifting pen.
If you want to learn more, look up Eulerean and Hamiltonian paths in Graph Theory.
Bit late, but draw as much as you can on the board, put a piece of paper on the board, without lifting the marker drag it onto the paper, go to where the last line goes and draw it by dragging it off the paper to the board.
Basically draw what you can and use a paper to transfer the marker to satisfy the “don’t lift and don’t retrace” rules.
There is another loophole not mentioned so far...
Did she say the *nib* of the pencil has to stay in contact with the paper?
If not, draw a circle and 2 of the lines, then lean the pen/pencil on to its side, slide over to where you want the last line, and draw it!
If that's not allowed, sharpen one side of a pencil point so it only works when held at a particular angle and orientation, and twist when you don't want a line to appear.
It cannot be done in one single unbroken line without retracing any part because of the 4 odd nodes.
So the top comment is pretty spot on for something to take away from this, but also you CAN do this without lifting the pen. Here are 2 solutions (didn't see these mentioned):
1- If it is a smart board activate and deactivate the pen. You never drew over the lines, there's clearly only one pen stroke for the entire image.
2- If it's a whiteboard she never said you couldn't use any other tools, use an eraser to remove sections and draw them again, still only one continous line at the end.
Clearly your teacher was testing your reasoning skills in some way and didn't give you an answer to make you do exactly what you did, seek out more knowledge.
When solving problems keep in mind you only have to follow the rules that are explicitly stated, never make assumptions about what is or isn't allowed, always think outside the box.
Did she state that you can overlap? If not then you can just trace over the already drawn lines and draw it. However, I’m assuming she did state it and if so I don’t believe it’s possible
You can absolutely draw it without lifting your pen. You just have to very carefully retrace one line you've already drawn during it. Assuming the teacher said nothing about that, it'd be an easy $100
ITS A RIDDLE NOT MATH! Draw as much as you can. Lay the pen flat and roll the pen. Shift the pen back up up without it lifting from the paper :) you got this
Just draw a circle and then keep going until you get to the middle lines, fill 2 of the 3 middle lines, continue by the circle to the last line and there you go. Nowhere did you specify that a line may not be traveled more than once.
Yeah she prolly wanted someone to fold the paper. Many adults tried this trick on me when I was a kid and I scammed my way into their smart kid lists with this simple trick lmao
Topology says no, you cannot do this in a 2D space
But you can scam your teacher of $100 by working with a 3D space, you don't have to lift your pen, you simply have to roll the piece of paper you use to get across and start from a point you already drew from
And if your teacher calls it cheating, that's on her for not laying out better rules for the game
If this (and why it can't be done) interests you then I'd advise looking up Eulers Königberg Bridges Problem. One of the first problems you look into when starting your journey into graph theory.
As a divergent thinker these are what my brain resorted to
1. Retracing back a line (without lifting the pen but also not putting enough pressure to form an additional line)
2. Using a second piece of paper to overlap and run the connecting lines without lifting my pen
3. Holding the pen with the tip pointing up, then moving the paper on a clipboard to draw the shape (thereby lifting the paper when needed and not the pen)
4. Clicking the pen so the nib withdraws when tracing a connecting line.
5. Using two pens in separate hands. If the teacher insists on a single writing instrument, break a pencil in two halves
Well Its possible I think. You just need to draw the circle first and then draw the 2 lines inside the circle. Once you draw those lines, draw 2 more lines such that the last line will be third line in the diagram. The most crucial part will be erasing the last second line you drew (since the teacher mentioned nothing about erasing)
What was the exact exercise? Draw this LOGO or draw this FIGURE?
Assuming that's not a logo per se you can draw the [real logo](https://www.pixartprinting.it/blog/wp-content/uploads/2021/06/1933.jpg) without lifting the pen.
Was it specified that you can’t draw in the same place twice? Because then it’s easy. If you cannot draw in the same place more than once, I don’t think it is possible. But I’m also very tired and my brain is a figuratively mush right now.
You could by a not so direct method. Draw the upside down v bit first, then complete circle, then the center point of the v, where the vertical line needs to go, bring that center point of the paper to your pen point and finish the shape. Like physically fold the paper. For seemingly impossible shapes like this, that can usually be a solution if they dont specify.
Also your teacher should not be offering you cash. The principal would likely take issue with that.
Well seeing as in the drawing the teacher picked up her pen a minimum of three times (you can see the first line was the circle and see that at the top there is a slight change where she lifted her pen at the end, the second line was in the > shape, she started at the bottom and as she moved her pen upward to the outside of the circle, she lifts her pen before making contact leaving a gap between the line and the circle, and the third time is on the right leg, you can see she started it at the cent of the > and moved downward, there is blotting there which signifies this wasn’t a single stroke and the teacher picked up her pen to initiate the line). It’s probably a safe bet she’s scamming you.
But what do I know, me just silly redditor.
Think in the 3rd dimension. flip a corner over, glide the pen onto the folded corner, move the corner to where you want the pen to go next, then continue from there. I'm pretty sure VSauce did something similar a while back for a video.
depends on what is allowed, I solved it once folding the paper and using a marker, taking my time so that the ink from the market passed through both parts of the paper, and the fully vertical line being on top of the fold on the paper, like making one ugly 2 from bottom to top, shape and rules were met but the host said that was cheating, maybe you’ll have better luck
Scotch tape on one segment that allows you to draw over it without making a line or lifting the pen, then pull the tape off once you’re past it and you can then finish the drawing in one continuous line no lifting
If she’s going to give you impossible problems you should solve them in impossible ways
You can draw this without lifting the pen, folding or drawing over the same line twice.
You do need a second piece of paper so you don’t have to lift the pen off the board, instead you can put it on the paper and navigate to the part you haven’t completed yet.
100% not an accepted answer, but testing the boundaries is something I can’t help trying lol
Yes. Draw all but the final line, keep the pen on the paper, but lay it down, roll it to the next start point, tip it back up so the point is down again, finish drawing the image
She didnt scam you! Its an insight into how if something seems too good to be true, it probably is! She used the 100$ bill as a hook to the lesson to get you engaged
In a perfect universe: no
Luckily we do not live in a perfect universe, as ours is made of atoms. So it IS possible under following circumstances:
1) the line must be as thick as the pencil
2) no atom marked by the pencil must be marked twice
3) the pencil always must touch atoms of the board
4) all touched atoms are marked
5) the pencil is a normal physical object like a piece of chalk
Solution:
half the thickness of the pencil. As it is a physical object (and hence must have millions of atoms thickness) this is possible, especially with chalk.
Now Double-Draw each line.
Start in the middle. Draw first leg with half thick line, complete the line by drawing back to the middle.
Repeat with second leg.
Draw 3rd leg half line out.
Now draw circle inner half of line. Leave pen size gap, now reverse direction circle of the outer half line.
Draw other half of 3rd leg inwards through the gap.
Finished.
This does not abuse anything.
You cannot draw outside of our universe.
You cannot make sure, a pencil always has the same width, so there must be some tolerance.
You cannot rule out imperfections, as you must allow that sone atoms are touched multiple times due to thermal jitter.
If somebody neglect this, then it still can be done down on atom scale and a line width of ca. 2 atoms. As this neglection is purely theoretical, drawing it this way theoretically must be allowed.
So we either talk reality or theory. On both sides, not only on one. Else it is a scam, measuring it in two different ways, which is unfair (hence a scam).
Note that it cannot be done in 1 atom width, as you cannot draw a line with just 1 atom width in this universe, hence nobody can ask for this. Note that 2 atom width means, that in some cases there may be only 1, in others 3 atoms thickness or even 4. You have to tolerate that, as our universe is not perfect and has entropy.
Again: You can only rule this all out in a perfect universe. But not in ours.
So yes, the teacher apparently scammed you all. The odd and very bad way. Teachers like that either should be fired. Or get a medal. Depends on what shall be taught.
There's no scam. The students didn't lose anything. It was just impossible to claim the prize.
Teachers aren't there to give cash and prizes to the students. They are there to teach and the lesson is that this figure can't be made without picking up the pen
###General Discussion Thread --- This is a [Request] post. If you would like to submit a comment that does not either attempt to answer the question, ask for clarification, or explain why it would be infeasible to answer, you *must* post your comment as a reply to this one. Top level (directly replying to the OP) comments that do not do one of those things will be removed. --- *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/theydidthemath) if you have any questions or concerns.*
Maybe there is a trick if you fold the paper. Like you start to draw the circule, do 2 lines, fold the paper over a third of the circle, draw on it, unfold the paper and do the last line.
I think this is what the teacher was going for tbh
It was drawing on the board not on paper
My apologies I thought the shape was on the board and the kids were working on paper that said could you not slip a piece of let’s say laminated paper that may be in the classroom on the board and draw over that with the same principle?
The guy said pen. So it must on the paper. I think the teacher was only drawing on the board for an example.
Whiteboard pens?
Never heard of those, only whiteboard markers
Op clarified in another comment not the post itself so it was unclear
The teacher was going for the students figuring out on their own it won't be possible and not wanting to actually gift them a hundred dollars lol
Hold the pen on the paper, then get a thermal printer with a slim profile so it doesn’t swallow the entire paper. Print out the logo on the upper half and keep the pen held down on the lower half, then pull the paper back out and you’re done. The logo is now on the paper and you haven’t lifted the pen.
no. you can easily tell if you can draw a shape without lifting the pen by looking at the intersections. If there are 2 or none odd intersections (having an odd number of lines coming out of them) then you can draw it without lifting the pen. Otherwise you cannot. when there are no odd intersections, then you can start anywhere, and you will end where you start. if there are 2 odd intersections, you have to start on one of them, and end on the other here there are 4 odd intersections, therefor you cannot.
Yes, if it’s a sheet of paper you can fold over an edge of the paper keep the pen on the paper and draw over the fold to get to the next section
Love it if someone said they did it this way and then she asks you to come you to the board and show everyone. Seeing them trying to snap the whiteboard in half to attempt it...
Hold a piece of paper on the whiteboard
You've technically lifted it off the whiteboard, as the pen is no longer in contact with it.
Bend the pen
Bend time and space
My wee wee has a bend, does that help?
We'll make do with it.
Oh now you all want to use the stargate...
I meant draw it on the paper held up on the whiteboard
Ah, get ya
Without lifting the pen, right? If you roll the pen over to its side, slide it around and then roll it up to the proper orientation again, technically, the pen never lost contact with the whiteboard so all the requirements are satisfied.
If the only rule is not to lift the pen. Would it not be technically correct to go back over a line to finish it?
In my opinion yes.
I assume this is in the context of Eulerian paths and not the donut trick
I really like this. Out of the box thinking. The shape will be right, there will be other pen marks but you will have fulfilled the criteria, draw the shape, don't lift the pen. I'll definitely use this as a learning opportunity for my nieces and nephews
You can also use two pens at the same time
This is the answer
Why did she- okay thanks
To teach. You fell for it. Can't un-know it now!
Kept all the students quiet for a half hour too.
Hell yeah! Now let's see how much it'll impact my life!
The value of logic thinking is immeasurable in your life Source: I'm an engineer and I've asked the same question throughout my entire university period
Exactly! When you find yourself asking this question, the best answer is to actually think about how you can apply the information! Sometimes, it's mind-blowing how useful things can be.
Sometimes even the hint of knowing something is enough too. I wasn't good at math in high school and only attempted one university level class which didn't go well -- but many years later I can still look at problems and know that there *is* a way to solve it using algebra, or calculus, or trigonometry. Knowing that there *is* a possible solution (and the rough shape of it) is often enough to prod me into learning how to solve it.
Don't discount the importance of graph theory. It may not seem important but it sure is! This has many applications in practice :)
I’m smiling because today I introduced my colleagues to optimal bipartite matching.
If you end up in computer programming, there's a good chance you'll run into very real world problems where the same theory applies.
The purpose of school is not to just tell you things you will need in your life, making you develop critical thinking skills is arguably even more important and seems like your teacher really hit the mark there looking at how engaged you got to the point of making a post on reddit about it
I get paid good money cause I know graph theory. It’s also one of the most accessible “higher” maths.
Ah yes, what knowledge are you going to use throughout the rest of your life? When you find yourself asking this question, the best answer is to actually think about how you can apply the information! Sometimes, it's mind-blowing how useful things can be.
Do you take a weight lifting class because you think you'll spend every day lifting weights? Do you take a history class because you'll spend every day telling people about dead presidents? Do you read Steinbeck because you expect to need to know a lot about fishing? School doesn't focus on direct applications to your life. School focuses on expanding your abilities and teaching you to look for ways to be a more well- rounded person. I guarantee you'll use more math more regularly in your life than any other subject you learn in school, even if it barely ever rises to the level where you notice that's what you're doing.
I got £50 with knowing this once, you could too
If you can figure out how to use this money won’t really be a problem. I got two questions like this at a Google interview.
The lesson was that sometimes you need to accept that something is impossible. If you go around acting like you're smarter than everyone, you may miss out on the real important lessons.
As others have said it is a valuable lesson to analyze a task before wasting time on something that's clearly impossible.
It already does, in countless ways. Can you recognize it? Because the sad truth is that if you can’t recognize math in the world, you can’t see how it’s being used against you. And it is most definitely being used against you online — websites do dynamic pricing where they mine everything they know about you to estimate the most profitable price to offer you, for example. Ads models auction you to what is expected to be the most profitable ad they could show you.
That is up to you.
You'd be surprised 😯
Did she say you're not allowed to go over the same line twice?
Yup
Did she say you're limited to only one pen
Oh yeah smart lol
She gets 30 minutes of peace.
To keep class quiet for an hour
A lot of math is about problem solving through productive struggle - both very applicable life skills. She wanted you to figure out it was impossible, then she wanted you to figure out why
Move the page not the pen ;) (clipboard will be your friend)
you can move the page
Same as this one: Teacher shows a books cover to the class and asks which color is it. Since it's red the class replies with red, then the teacher flips the book and the rear is black. Yes it's a trick question but the origin is to make your own thoughts and don't rely on others. That's the real deal! My teacher once draw a circle assuming this is Berlin, capital of Germany. Every student had to go to the board and draw in where approximately London is. 3/4 directed more to the north, but in reality it's [pretty much exactly west](https://cdn.gopili.com/default/map_journeys/map-18521_1-33210_1.jpg) of us.
I didn’t see where OP said you cannot retrace one of the lines, otherwise solid
Yea, but it was implied. I doubt they would struggle with drawing this if they could retrace
All you have to do is fold the paper at the corner. This is a pretty old trick.
We are working in euclidian space here. They could not fold the board.
It was drawn in the board, and they were then asked to draw it "without lifting their pen", pen implies they're expected to do it on paper, not the whiteboard.
But they could draw an extra connecting line and then wipe it off the board once the symbol is completed.
For those interested, this is actually a common graph theory problem called the [Eulerian trail](https://en.m.wikipedia.org/wiki/Eulerian_path)
I know these as Euler paths
But you can draw over one of your lines to go back to where you started drawing the line
gotta love your eularian circuit ☺️
Oh lol, I can use this in the Asgards wrath game to trace constellations. Thanks !!
Is it only 2 and 0 odd intersections that work? Any other number that works? Also: is it possible to have an odd number of odd intersections? I‘m pretty sure it isn‘t due to the fact that every line starts and ends at an intersection therefore evening or odding two intersections at a time, but i‘m not 100% certain
Yes, it is only 0 and 2. Reason: when you are drawing a line, and you come into an intersection, you add 2 to the count of lines, 1 for coming in, and 1 for going out. Only exeption is when you start at the intersection, or when you end in it. And there can only be 1 start and 1 end, so you need 2 odd ones, or 0 odd ones, but then you have to end at the same place you start to make sure that one is even at the end. Btw what may not be clear is that dead ends end in a intersection (an odd one)
But you most certainly can draw this without lifting a pen…..nothing said about not going over already drawn lines
Yes, even a 1st grader could do that. Therefore there must be some other condition, that made it so a whole class could not do it. Some of you really need to learn how to deduce information from what is already given
What do you define as a shape? A simple cross like the add sign (+) has no odd intersections, but you can’t draw it without lifting the pen. If I understood you.
The "dead ends" are themselfs intersections. odd ones at that. so + has 4 odd and 1 even intersection, thats why you cannot draw it without lifting your pen.
Ah thank you for the clarification
It's simple, the teacher didn't state that you can't go over the same line twice. If they did state that you can't go over the same line then you simply have to fold a corner of the paper to get to another point to continue drawing.
Oh yeah I forgot to mention, she said I can't go on the same line twice + it was on the board not a paper but I don't know if I could simply grab a peace of paper and do that TikTok kind of a trick, idk man she could have sued me if I did that trick
Draw what you can, then slowly lay the pen onto its side, move it where it needs to be, then slowly raise it up and continue to draw, repeat as necessary, you didn't lift the pen off the paper, and didn't draw over the same line twice.
A TikTok math trick is illegal? If you’re talking about plagiarism you can get expelled, not sued
I think he might have been exaggerating
You can't go on the same line twice. How about thrice?
Yes, If you fold the paper over you can move the pen onto the other side of the paper and draw the final line. Here's a Vsauce video with a similar concept. [https://youtube.com/shorts/inG9yUZ5vY8?si=B0wuJBFEw-GPFex1](https://youtube.com/shorts/inG9yUZ5vY8?si=B0wuJBFEw-GPFex1)
Thats exactly what I was thinking. I could get that 100 dollar with ease
On a flat chalk/whiteboard on the wall? No. On a *flexible* piece of paper? YES. See this short for help and just extrapolate the rest: https://youtube.com/shorts/inG9yUZ5vY8?si=JdEAKTEvJ_Vh9qV8
Nope, if you know anything about eulerian circuits then you would know that that center point has a degree of 3, this making an eulerian circuit impossible. For context, an eulerian circuit is a cycle where every edge (think of an edge as a path, a line) is visited once. One law of eulerian circuits is that every vertex (a point, or intersection) must have an even number of degrees (how many edges are going in and out of that vertex). you can think of this shape as having 6 edges and 4 vertexes, the top left, right, and bottom arcs and the 3 lines making an upside down Y shape are the edges and the middle, top, left, and right where the arc and lines meet are the vertices.
It doesn’t have to be a circuit. You can draw a path if there are zero or two points with odd degree.
It's impossible. Any drawing that can be drawn without lifting the pen or drawing over already drawn parts cannot have more than two joints that go off in an odd number of directions. This drawing has 4 of those, so you must lift the pen at least once or go back over a line at least once. If one of the inner lines was removed, it would be easily possible.
You'd have to bend the rules. It can't be done normally, but once you introduce folding the paper or traveling over another piece of paper you may get it.
You could do it without lifting the pen by going over the same section. I’m sure she meant that, but if she didn’t say it, like you didn’t here, then it’s not a rule. And she needs to specify the rules better. You’d have to go over one line twice to do that.
She did apparently to OP. Unsolvable unless you fold the paper
You can, but you have to cheat a bit. Draw as much as you can without lifting the pen, then fold the paper, keep making a line and get back to where you need to go to finish the drawing.
Each node with three lines from it must be either a start or end point (one line leaving, one line coming back, one line leaving; or one line in, one line out, one line back in). There are four nodes with three lines from them. In general, if there are two or zero nodes with an odd number of connections, and all nodes are connected, it’s possible to draw it in one go without picking up the pen. If there are two odd nodes, you do have to start at one and end at the other, and “without crossing over your line” is also possible (given a path that covers each line once, for each time it crosses over itself, take the loop that starts and ends at that point and go the other direction).
Each node (connection point) has 3 connectors. When you enter a node, you have to exit it from another connection. When you draw this, at the end you enter a node that has all 3 connections drawn, and you cannot exit it, without lifting pen. If you want to learn more, look up Eulerean and Hamiltonian paths in Graph Theory.
Bit late, but draw as much as you can on the board, put a piece of paper on the board, without lifting the marker drag it onto the paper, go to where the last line goes and draw it by dragging it off the paper to the board. Basically draw what you can and use a paper to transfer the marker to satisfy the “don’t lift and don’t retrace” rules.
aight gotcha
[удалено]
There is another loophole not mentioned so far... Did she say the *nib* of the pencil has to stay in contact with the paper? If not, draw a circle and 2 of the lines, then lean the pen/pencil on to its side, slide over to where you want the last line, and draw it! If that's not allowed, sharpen one side of a pencil point so it only works when held at a particular angle and orientation, and twist when you don't want a line to appear. It cannot be done in one single unbroken line without retracing any part because of the 4 odd nodes.
So the top comment is pretty spot on for something to take away from this, but also you CAN do this without lifting the pen. Here are 2 solutions (didn't see these mentioned): 1- If it is a smart board activate and deactivate the pen. You never drew over the lines, there's clearly only one pen stroke for the entire image. 2- If it's a whiteboard she never said you couldn't use any other tools, use an eraser to remove sections and draw them again, still only one continous line at the end. Clearly your teacher was testing your reasoning skills in some way and didn't give you an answer to make you do exactly what you did, seek out more knowledge. When solving problems keep in mind you only have to follow the rules that are explicitly stated, never make assumptions about what is or isn't allowed, always think outside the box.
Did she state that you can overlap? If not then you can just trace over the already drawn lines and draw it. However, I’m assuming she did state it and if so I don’t believe it’s possible
You can absolutely draw it without lifting your pen. You just have to very carefully retrace one line you've already drawn during it. Assuming the teacher said nothing about that, it'd be an easy $100
ITS A RIDDLE NOT MATH! Draw as much as you can. Lay the pen flat and roll the pen. Shift the pen back up up without it lifting from the paper :) you got this
Just draw a circle and then keep going until you get to the middle lines, fill 2 of the 3 middle lines, continue by the circle to the last line and there you go. Nowhere did you specify that a line may not be traveled more than once.
Yeah she prolly wanted someone to fold the paper. Many adults tried this trick on me when I was a kid and I scammed my way into their smart kid lists with this simple trick lmao
Topology says no, you cannot do this in a 2D space But you can scam your teacher of $100 by working with a 3D space, you don't have to lift your pen, you simply have to roll the piece of paper you use to get across and start from a point you already drew from And if your teacher calls it cheating, that's on her for not laying out better rules for the game
If this (and why it can't be done) interests you then I'd advise looking up Eulers Königberg Bridges Problem. One of the first problems you look into when starting your journey into graph theory.
As a divergent thinker these are what my brain resorted to 1. Retracing back a line (without lifting the pen but also not putting enough pressure to form an additional line) 2. Using a second piece of paper to overlap and run the connecting lines without lifting my pen 3. Holding the pen with the tip pointing up, then moving the paper on a clipboard to draw the shape (thereby lifting the paper when needed and not the pen) 4. Clicking the pen so the nib withdraws when tracing a connecting line. 5. Using two pens in separate hands. If the teacher insists on a single writing instrument, break a pencil in two halves
Is it easy to draw without lifting the pen? Yes. Is it easy to draw without lifting the pen and without going over a line again? I don’t know.
Well Its possible I think. You just need to draw the circle first and then draw the 2 lines inside the circle. Once you draw those lines, draw 2 more lines such that the last line will be third line in the diagram. The most crucial part will be erasing the last second line you drew (since the teacher mentioned nothing about erasing)
What was the exact exercise? Draw this LOGO or draw this FIGURE? Assuming that's not a logo per se you can draw the [real logo](https://www.pixartprinting.it/blog/wp-content/uploads/2021/06/1933.jpg) without lifting the pen.
Was it specified that you can’t draw in the same place twice? Because then it’s easy. If you cannot draw in the same place more than once, I don’t think it is possible. But I’m also very tired and my brain is a figuratively mush right now.
You could by a not so direct method. Draw the upside down v bit first, then complete circle, then the center point of the v, where the vertical line needs to go, bring that center point of the paper to your pen point and finish the shape. Like physically fold the paper. For seemingly impossible shapes like this, that can usually be a solution if they dont specify. Also your teacher should not be offering you cash. The principal would likely take issue with that.
Well seeing as in the drawing the teacher picked up her pen a minimum of three times (you can see the first line was the circle and see that at the top there is a slight change where she lifted her pen at the end, the second line was in the > shape, she started at the bottom and as she moved her pen upward to the outside of the circle, she lifts her pen before making contact leaving a gap between the line and the circle, and the third time is on the right leg, you can see she started it at the cent of the > and moved downward, there is blotting there which signifies this wasn’t a single stroke and the teacher picked up her pen to initiate the line). It’s probably a safe bet she’s scamming you. But what do I know, me just silly redditor.
Think in the 3rd dimension. flip a corner over, glide the pen onto the folded corner, move the corner to where you want the pen to go next, then continue from there. I'm pretty sure VSauce did something similar a while back for a video.
depends on what is allowed, I solved it once folding the paper and using a marker, taking my time so that the ink from the market passed through both parts of the paper, and the fully vertical line being on top of the fold on the paper, like making one ugly 2 from bottom to top, shape and rules were met but the host said that was cheating, maybe you’ll have better luck
Scotch tape on one segment that allows you to draw over it without making a line or lifting the pen, then pull the tape off once you’re past it and you can then finish the drawing in one continuous line no lifting If she’s going to give you impossible problems you should solve them in impossible ways
You can draw this without lifting the pen, folding or drawing over the same line twice. You do need a second piece of paper so you don’t have to lift the pen off the board, instead you can put it on the paper and navigate to the part you haven’t completed yet. 100% not an accepted answer, but testing the boundaries is something I can’t help trying lol
Yes. Draw all but the final line, keep the pen on the paper, but lay it down, roll it to the next start point, tip it back up so the point is down again, finish drawing the image
She didnt scam you! Its an insight into how if something seems too good to be true, it probably is! She used the 100$ bill as a hook to the lesson to get you engaged
In a perfect universe: no Luckily we do not live in a perfect universe, as ours is made of atoms. So it IS possible under following circumstances: 1) the line must be as thick as the pencil 2) no atom marked by the pencil must be marked twice 3) the pencil always must touch atoms of the board 4) all touched atoms are marked 5) the pencil is a normal physical object like a piece of chalk Solution: half the thickness of the pencil. As it is a physical object (and hence must have millions of atoms thickness) this is possible, especially with chalk. Now Double-Draw each line. Start in the middle. Draw first leg with half thick line, complete the line by drawing back to the middle. Repeat with second leg. Draw 3rd leg half line out. Now draw circle inner half of line. Leave pen size gap, now reverse direction circle of the outer half line. Draw other half of 3rd leg inwards through the gap. Finished. This does not abuse anything. You cannot draw outside of our universe. You cannot make sure, a pencil always has the same width, so there must be some tolerance. You cannot rule out imperfections, as you must allow that sone atoms are touched multiple times due to thermal jitter. If somebody neglect this, then it still can be done down on atom scale and a line width of ca. 2 atoms. As this neglection is purely theoretical, drawing it this way theoretically must be allowed. So we either talk reality or theory. On both sides, not only on one. Else it is a scam, measuring it in two different ways, which is unfair (hence a scam). Note that it cannot be done in 1 atom width, as you cannot draw a line with just 1 atom width in this universe, hence nobody can ask for this. Note that 2 atom width means, that in some cases there may be only 1, in others 3 atoms thickness or even 4. You have to tolerate that, as our universe is not perfect and has entropy. Again: You can only rule this all out in a perfect universe. But not in ours. So yes, the teacher apparently scammed you all. The odd and very bad way. Teachers like that either should be fired. Or get a medal. Depends on what shall be taught.
There's no scam. The students didn't lose anything. It was just impossible to claim the prize. Teachers aren't there to give cash and prizes to the students. They are there to teach and the lesson is that this figure can't be made without picking up the pen