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Connecting Colored Boxes on the Plane

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Place any number of colored boxes on the infinite plane. You have to connect similarly colored boxes, so boxes of each color form a single cycle (each colored box is connected to two similarly colored boxes).

Given that boxes do not touch each other, prove that it is always possible to connect the boxes in this manner such that there are no intersections between the connecting lines.

Inspired by Connecting Colored Boxes. However, that problem is different in nature because it is contained in a finite plane and some boxes are on the edge of that plane.

Note by Daniel Liu
3 years, 2 months ago

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Connect the boxes of the same color in a cycle (in any order). Do it for all colors. Then all nodes have degree \(2\). Hence by Kuratowski's theorem the resulting graph is planar. Abhishek Sinha · 3 years, 2 months ago

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@Abhishek Sinha Hey Hello! Archiet Dev · 3 years, 1 month ago

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@Abhishek Sinha Hey! I want to know something about you....How did you get to MIT ?means by giving SAT or something else... Archiet Dev · 3 years, 1 month ago

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Hey Daniel Liu! your post looksgood as always but oftenly your image does'nt read correctly on my phone, the picture did'nt show up. So try to change the web you are saving those picture, i think it's.. not only happen to me. Hafizh Ahsan Permana · 3 years, 2 months ago

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@Hafizh Ahsan Permana Chances are, if the image isn't reading, then it's a problem on your phone or on brilliant's phone app. Sorry for the inconvenience :\ Daniel Liu · 3 years, 2 months ago

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