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garden

A gardener wants to trisect his rectangular garden into three parts, with the proviso that he can neither enter on the left and leave on the right AND nor can he enter at the top and leave at the bottom.

But he can't seem to do it! Can you?

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  • $\begingroup$ Do the 3 parts need to be equal in area? $\endgroup$
    – Stiv
    Dec 24, 2019 at 6:52
  • $\begingroup$ No, that's just my picture! $\endgroup$
    – JMP
    Dec 24, 2019 at 6:55
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    $\begingroup$ To clarify: will the farmer walk through one of the three green sections without crossing the black lines, or will he walk along the black lines without stepping into the green? $\endgroup$
    – ZanyG
    Dec 24, 2019 at 7:27
  • $\begingroup$ The lines in my picture are the walls, and the gardener can go from left to right in the bottom garden. $\endgroup$
    – JMP
    Dec 24, 2019 at 7:29
  • $\begingroup$ I believe I have a proof that this is impossible, if I correctly understand what you are asking. Are you sure that there is a solution? $\endgroup$
    – ZanyG
    Dec 24, 2019 at 7:37

2 Answers 2

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I suspect

that this is impossible.

Reasoning:

Each of the four corners must belong to one of the three sections. By the pigeonhole principle, at least one section must own at least two corners, and since that section is a traversable region, there exists a path between those two corners. However, this is disallowed, since it will traverse either the entire horizontal length or vertical with (or both) of the field. The trisection is thus impossible.

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  • $\begingroup$ Exactly what I got! $\endgroup$
    – JMP
    Dec 24, 2019 at 7:45
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Why not?

If he trisects the garden like a flag of the Czech republic (depicted below) (by color), all conditions will be satisfied (note that white-blue and red-blue borders go exactly to the corners). enter image description here

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  • $\begingroup$ Right on the left border, or left to right of the top and base seem to contradict that. $\endgroup$
    – JMP
    Dec 24, 2019 at 6:58
  • $\begingroup$ @JMP What do you mean? $\endgroup$
    – trolley813
    Dec 24, 2019 at 6:59
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    $\begingroup$ ^ @JMP But that would assume your wall has no thickness at all. In reality it will be a number of bricks wide - I have a few solutions that work in the same way as trolley813's when you can assume the wall separates the field entirely at corners, but I don't understand why this is invalid from the specification? Thanks. $\endgroup$
    – Stiv
    Dec 24, 2019 at 7:32
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    $\begingroup$ @Stiv Actually even if the wall has zero thickness, the points under the wall belong to the no part of the garden. $\endgroup$
    – trolley813
    Dec 24, 2019 at 7:35
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    $\begingroup$ @JMP that looks rather a lame riddle then. I see the geometry tag, but in the post you describe the problem as down to earth as conceivably possible. Well, down to earth gardens have walls of appreciable thickness, same as the actual people who walk in the gardens. $\endgroup$
    – Gnudiff
    Dec 24, 2019 at 21:21

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