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enter image description here 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?

Blue
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1Spectre1
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    https://en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg – Martin R Nov 23 '18 at 11:39
  • Similar: https://math.stackexchange.com/questions/292909/proof-cannot-draw-this-figure-without-lifting-the-pen – Martin R Nov 23 '18 at 11:46
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    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. –  Nov 23 '18 at 11:46
  • @Rahul Your argument is circular, you need to prove why you cannot draw any figure you like without assuming it. – Alex S Feb 07 '19 at 00:43

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No, as there are four 5-nodes.

Richard Martin
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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.

Bram28
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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.

Atharva Kathale
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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.

MBorg
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