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Problem 12.3: Three unfriendly neighbors use the same water, oil and treacle wells. In order to avoid meeting, they wish to build non-crossing paths from each of their houses to each of the three wells. Can this be done?

We know that the answer of this question is No, since the bipartite graph $k_{3,3}$ is not a planar graph.

My Question: Are there some questions such as Problem 12.3 that ask to do or draw something but mathematically is not possible.

I would appreciate to suggest questions with math level similar to Problem 12.3.

Please edit my question for suitable tags. Thanks

Amin235
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    Possible duplicate of Math problems that are impossible to solve – Parcly Taxel Feb 18 '18 at 11:34
  • @ParclyTaxel I know my question is similar to that question you mentioned but my aim to ask this question is that to ask some question that its appearance is not mathematically and has no answer. Thanks. – Amin235 Feb 18 '18 at 11:38
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    I want to hear more about the treacle well. Do these neighbors also have access to a flapjack tree? – MJD Feb 18 '18 at 12:21
  • One possible solution to this seemingly insoluble problem is for the neighbors to live on a toroidal surface; then there is no problem embedding a $K_{3,3}$. Normally, this suggestion is far-fetched. But if they have a treacle well, perhaps it's because they are actually living on a giant donut! – MJD Feb 18 '18 at 15:52
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    I'm very sorry if I made you feel uncomfortable with my silly comments. They were not intended to mock you. I only found the idea of a treacle well delightful! – MJD Feb 18 '18 at 21:44
  • @MJD No problem. I am one of the follower of your blog. I like your writing. Thanks for your comments. – Amin235 Feb 19 '18 at 08:01

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After your comment, I want to add something but it will be too long for a comment.

In graph theory, not all the questions are stated in this way (like giving an example from real life) but questions that ask "Prove that the following graph is non-planar" are all the same kind and although it is not possible, we can prove that it is impossible to draw those graphs planarly (For example see non-planarity of Petersen Graph using Kuratowski's theorem).

As given in the duplicate, there are also some questions that ask whether "Euler Path" exists or not as in the problem Seven Bridges of Königsberg, which can be proven that such path doesn't exist.

However these questions are different from the questions that have "no answer". In mathematics, generally proving impossibility of a statement is as valuable or important as proving that statement holds. Because question asks "Can this be done?" and answer is simply "No" and it can be proven when this is the case.

However in an example like Travelling Salesman Problem, the question "Can it be done with a certain complexity" is an open question and has no known answer (Notice that it is a matter of complexity so we know that it can be solved in some complexity but we don't know whether there exists an algorithm that solves it more efficiently).

ArsenBerk
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