Drawing a figure without lifting the pen
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1
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A friend of mine said that if you can draw the figure below without lifting your hand and without going over a piece for a second time, you can do whatever you want. I couldn't do it, no matter how I tried. There's always one piece missing.
Is it possible to draw this shape without lifting the pen and passing over the part already drawn?
puzzle
edited Nov 23 at 12:03
Blue
47k870148
asked Nov 23 at 11:28
1Spectre1
818
add a comment |
up vote
1
down vote
favorite
A friend of mine said that if you can draw the figure below without lifting your hand and without going over a piece for a second time, you can do whatever you want. I couldn't do it, no matter how I tried. There's always one piece missing.
Is it possible to draw this shape without lifting the pen and passing over the part already drawn?
puzzle
edited Nov 23 at 12:03
Blue
47k870148
asked Nov 23 at 11:28
1Spectre1
818
2
en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg
– Martin R
Nov 23 at 11:39
Similar: math.stackexchange.com/questions/292909/…
– Martin R
Nov 23 at 11:46
1
Proof by contradiction: Suppose you can draw the figure without lifting your hand and without going over a segment twice. It is given in the problem that if this is possible, then you can do whatever you want. However, we already know that you cannot do whatever you want. Therefore, the assumption must be false.
– Rahul
Nov 23 at 11:46
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
A friend of mine said that if you can draw the figure below without lifting your hand and without going over a piece for a second time, you can do whatever you want. I couldn't do it, no matter how I tried. There's always one piece missing.
Is it possible to draw this shape without lifting the pen and passing over the part already drawn?
puzzle
edited Nov 23 at 12:03
Blue
47k870148
asked Nov 23 at 11:28
1Spectre1
818
A friend of mine said that if you can draw the figure below without lifting your hand and without going over a piece for a second time, you can do whatever you want. I couldn't do it, no matter how I tried. There's always one piece missing.
Is it possible to draw this shape without lifting the pen and passing over the part already drawn?
puzzle
puzzle
edited Nov 23 at 12:03
Blue
47k870148
asked Nov 23 at 11:28
1Spectre1
818
edited Nov 23 at 12:03
Blue
47k870148
asked Nov 23 at 11:28
1Spectre1
818
edited Nov 23 at 12:03
Blue
47k870148
edited Nov 23 at 12:03
Blue
47k870148
edited Nov 23 at 12:03
Blue
47k870148
47k870148
asked Nov 23 at 11:28
1Spectre1
818
asked Nov 23 at 11:28
1Spectre1
818
asked Nov 23 at 11:28
1Spectre1
818
818
2
en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg
– Martin R
Nov 23 at 11:39
Similar: math.stackexchange.com/questions/292909/…
– Martin R
Nov 23 at 11:46
1
Proof by contradiction: Suppose you can draw the figure without lifting your hand and without going over a segment twice. It is given in the problem that if this is possible, then you can do whatever you want. However, we already know that you cannot do whatever you want. Therefore, the assumption must be false.
– Rahul
Nov 23 at 11:46
add a comment |
2
en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg
– Martin R
Nov 23 at 11:39
Similar: math.stackexchange.com/questions/292909/…
– Martin R
Nov 23 at 11:46
1
Proof by contradiction: Suppose you can draw the figure without lifting your hand and without going over a segment twice. It is given in the problem that if this is possible, then you can do whatever you want. However, we already know that you cannot do whatever you want. Therefore, the assumption must be false.
– Rahul
Nov 23 at 11:46
2
2
en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg
– Martin R
Nov 23 at 11:39
en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg
– Martin R
Nov 23 at 11:39
Similar: math.stackexchange.com/questions/292909/…
– Martin R
Nov 23 at 11:46
Similar: math.stackexchange.com/questions/292909/…
– Martin R
Nov 23 at 11:46
1
1
Proof by contradiction: Suppose you can draw the figure without lifting your hand and without going over a segment twice. It is given in the problem that if this is possible, then you can do whatever you want. However, we already know that you cannot do whatever you want. Therefore, the assumption must be false.
– Rahul
Nov 23 at 11:46
Proof by contradiction: Suppose you can draw the figure without lifting your hand and without going over a segment twice. It is given in the problem that if this is possible, then you can do whatever you want. However, we already know that you cannot do whatever you want. Therefore, the assumption must be false.
– Rahul
Nov 23 at 11:46
add a comment |
4 Answers
4
active
oldest
votes
up vote
3
down vote
accepted
It cannot be done and here is why. Let’s call the points where several lines together intersections. Now, if you had a single path that would cover the whole figure without going over any of the lines twice, then that means that any time you go into an intersection drawing one line, you need to leave that intersection again using a different line. This means that every intersection should be an intersection of an even number of lines. The only possible exceptions to this is when you start the path or end the path, so two intersections can have an odd number of lines ... but the rest must be all even. However, note that you have four intersections with five lines. So, it cannot be done.
answered Nov 23 at 13:42
Bram28
58.7k44185
add a comment |
up vote
3
down vote
No, as there are four 5-nodes.
answered Nov 23 at 11:32
Richard Martin
1,6328
add a comment |
up vote
0
down vote
We'll as just it is not mentioned in the question that we have to use a single hand. Try using 2 hands and draw.
answered Nov 26 at 8:57
Atharva Kathale
789
add a comment |
up vote
0
down vote
He could be asking a trick question. You could draw that figure with your finger on say an iPad, whilst keeping your hand rested the entire tea (but still lifting your finger). If you're drawing it in such a method, then you could do whatever you want with that image. Not just not going over a piece for a second time (by lightning your finger and keeping hand rested once again), but also add extra lines, make it red, delete it entirely. In that case, what your friend said is true - you can draw it without lifting your hand and do whatever you want.
In terms of solving it the proper mathematical way by only using a pen and no tricks like in the above paragraph, then no you cannot. As alluded to in the other answers, the number of nodes (points where several lines intersect) connected to an odd number of lines cannot exceed 2. In this case there are 4 odd-numbered nodes (nodes connected to an odd number of lines), in this case the corners of the inner rectangle. The reason behind this is that to cross every line without lifting your pen, each odd-numbered node must either be started at or ended at (or both). That is not possible if there are more than 2 odd-numbered nodes.
answered Nov 27 at 9:37
MBorg
1751114
add a comment |
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
It cannot be done and here is why. Let’s call the points where several lines together intersections. Now, if you had a single path that would cover the whole figure without going over any of the lines twice, then that means that any time you go into an intersection drawing one line, you need to leave that intersection again using a different line. This means that every intersection should be an intersection of an even number of lines. The only possible exceptions to this is when you start the path or end the path, so two intersections can have an odd number of lines ... but the rest must be all even. However, note that you have four intersections with five lines. So, it cannot be done.
answered Nov 23 at 13:42
Bram28
58.7k44185
add a comment |
up vote
3
down vote
accepted
It cannot be done and here is why. Let’s call the points where several lines together intersections. Now, if you had a single path that would cover the whole figure without going over any of the lines twice, then that means that any time you go into an intersection drawing one line, you need to leave that intersection again using a different line. This means that every intersection should be an intersection of an even number of lines. The only possible exceptions to this is when you start the path or end the path, so two intersections can have an odd number of lines ... but the rest must be all even. However, note that you have four intersections with five lines. So, it cannot be done.
answered Nov 23 at 13:42
Bram28
58.7k44185
add a comment |
up vote
3
down vote
accepted
up vote
3
down vote
accepted
It cannot be done and here is why. Let’s call the points where several lines together intersections. Now, if you had a single path that would cover the whole figure without going over any of the lines twice, then that means that any time you go into an intersection drawing one line, you need to leave that intersection again using a different line. This means that every intersection should be an intersection of an even number of lines. The only possible exceptions to this is when you start the path or end the path, so two intersections can have an odd number of lines ... but the rest must be all even. However, note that you have four intersections with five lines. So, it cannot be done.
answered Nov 23 at 13:42
Bram28
58.7k44185
It cannot be done and here is why. Let’s call the points where several lines together intersections. Now, if you had a single path that would cover the whole figure without going over any of the lines twice, then that means that any time you go into an intersection drawing one line, you need to leave that intersection again using a different line. This means that every intersection should be an intersection of an even number of lines. The only possible exceptions to this is when you start the path or end the path, so two intersections can have an odd number of lines ... but the rest must be all even. However, note that you have four intersections with five lines. So, it cannot be done.
answered Nov 23 at 13:42
Bram28
58.7k44185
edited Nov 23 at 18:16
answered Nov 23 at 13:42
Bram28
58.7k44185
answered Nov 23 at 13:42
Bram28
58.7k44185
answered Nov 23 at 13:42
Bram28
58.7k44185
58.7k44185
add a comment |
add a comment |
up vote
3
down vote
No, as there are four 5-nodes.
answered Nov 23 at 11:32
Richard Martin
1,6328
add a comment |
up vote
3
down vote
No, as there are four 5-nodes.
answered Nov 23 at 11:32
Richard Martin
1,6328
add a comment |
up vote
3
down vote
up vote
3
down vote
No, as there are four 5-nodes.
answered Nov 23 at 11:32
Richard Martin
1,6328
No, as there are four 5-nodes.
answered Nov 23 at 11:32
Richard Martin
1,6328
answered Nov 23 at 11:32
Richard Martin
1,6328
answered Nov 23 at 11:32
Richard Martin
1,6328
answered Nov 23 at 11:32
Richard Martin
1,6328
1,6328
add a comment |
add a comment |
up vote
0
down vote
We'll as just it is not mentioned in the question that we have to use a single hand. Try using 2 hands and draw.
answered Nov 26 at 8:57
Atharva Kathale
789
add a comment |
up vote
0
down vote
We'll as just it is not mentioned in the question that we have to use a single hand. Try using 2 hands and draw.
answered Nov 26 at 8:57
Atharva Kathale
789
add a comment |
up vote
0
down vote
up vote
0
down vote
We'll as just it is not mentioned in the question that we have to use a single hand. Try using 2 hands and draw.
answered Nov 26 at 8:57
Atharva Kathale
789
We'll as just it is not mentioned in the question that we have to use a single hand. Try using 2 hands and draw.
answered Nov 26 at 8:57
Atharva Kathale
789
answered Nov 26 at 8:57
Atharva Kathale
789
answered Nov 26 at 8:57
Atharva Kathale
789
answered Nov 26 at 8:57
Atharva Kathale
789
789
add a comment |
add a comment |
up vote
0
down vote
He could be asking a trick question. You could draw that figure with your finger on say an iPad, whilst keeping your hand rested the entire tea (but still lifting your finger). If you're drawing it in such a method, then you could do whatever you want with that image. Not just not going over a piece for a second time (by lightning your finger and keeping hand rested once again), but also add extra lines, make it red, delete it entirely. In that case, what your friend said is true - you can draw it without lifting your hand and do whatever you want.
In terms of solving it the proper mathematical way by only using a pen and no tricks like in the above paragraph, then no you cannot. As alluded to in the other answers, the number of nodes (points where several lines intersect) connected to an odd number of lines cannot exceed 2. In this case there are 4 odd-numbered nodes (nodes connected to an odd number of lines), in this case the corners of the inner rectangle. The reason behind this is that to cross every line without lifting your pen, each odd-numbered node must either be started at or ended at (or both). That is not possible if there are more than 2 odd-numbered nodes.
answered Nov 27 at 9:37
MBorg
1751114
add a comment |
up vote
0
down vote
He could be asking a trick question. You could draw that figure with your finger on say an iPad, whilst keeping your hand rested the entire tea (but still lifting your finger). If you're drawing it in such a method, then you could do whatever you want with that image. Not just not going over a piece for a second time (by lightning your finger and keeping hand rested once again), but also add extra lines, make it red, delete it entirely. In that case, what your friend said is true - you can draw it without lifting your hand and do whatever you want.
In terms of solving it the proper mathematical way by only using a pen and no tricks like in the above paragraph, then no you cannot. As alluded to in the other answers, the number of nodes (points where several lines intersect) connected to an odd number of lines cannot exceed 2. In this case there are 4 odd-numbered nodes (nodes connected to an odd number of lines), in this case the corners of the inner rectangle. The reason behind this is that to cross every line without lifting your pen, each odd-numbered node must either be started at or ended at (or both). That is not possible if there are more than 2 odd-numbered nodes.
answered Nov 27 at 9:37
MBorg
1751114
add a comment |
up vote
0
down vote
up vote
0
down vote
He could be asking a trick question. You could draw that figure with your finger on say an iPad, whilst keeping your hand rested the entire tea (but still lifting your finger). If you're drawing it in such a method, then you could do whatever you want with that image. Not just not going over a piece for a second time (by lightning your finger and keeping hand rested once again), but also add extra lines, make it red, delete it entirely. In that case, what your friend said is true - you can draw it without lifting your hand and do whatever you want.
In terms of solving it the proper mathematical way by only using a pen and no tricks like in the above paragraph, then no you cannot. As alluded to in the other answers, the number of nodes (points where several lines intersect) connected to an odd number of lines cannot exceed 2. In this case there are 4 odd-numbered nodes (nodes connected to an odd number of lines), in this case the corners of the inner rectangle. The reason behind this is that to cross every line without lifting your pen, each odd-numbered node must either be started at or ended at (or both). That is not possible if there are more than 2 odd-numbered nodes.
answered Nov 27 at 9:37
MBorg
1751114
He could be asking a trick question. You could draw that figure with your finger on say an iPad, whilst keeping your hand rested the entire tea (but still lifting your finger). If you're drawing it in such a method, then you could do whatever you want with that image. Not just not going over a piece for a second time (by lightning your finger and keeping hand rested once again), but also add extra lines, make it red, delete it entirely. In that case, what your friend said is true - you can draw it without lifting your hand and do whatever you want.
In terms of solving it the proper mathematical way by only using a pen and no tricks like in the above paragraph, then no you cannot. As alluded to in the other answers, the number of nodes (points where several lines intersect) connected to an odd number of lines cannot exceed 2. In this case there are 4 odd-numbered nodes (nodes connected to an odd number of lines), in this case the corners of the inner rectangle. The reason behind this is that to cross every line without lifting your pen, each odd-numbered node must either be started at or ended at (or both). That is not possible if there are more than 2 odd-numbered nodes.
answered Nov 27 at 9:37
MBorg
1751114
answered Nov 27 at 9:37
MBorg
1751114
answered Nov 27 at 9:37
MBorg
1751114
answered Nov 27 at 9:37
MBorg
1751114
1751114
add a comment |
add a comment |
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2
en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg
– Martin R
Nov 23 at 11:39
Similar: math.stackexchange.com/questions/292909/…
– Martin R
Nov 23 at 11:46
1
Proof by contradiction: Suppose you can draw the figure without lifting your hand and without going over a segment twice. It is given in the problem that if this is possible, then you can do whatever you want. However, we already know that you cannot do whatever you want. Therefore, the assumption must be false.
– Rahul
Nov 23 at 11:46