Proof without words that the three utilities problem has no solution by CMG Lee. The leftmost house is first deleted. The lines connecting the utilities with the remaining houses divide the plane into three regions, shaded yellow, red and blue. Whichever region the deleted house is placed into, one of the utilities is outside the region. The Jordan curve theorem states that a line connecting them must intersect one of the existing lines.
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{{Information |Description=Proof without words that the three utilities problem has no solution by CMG Lee. The three dashed lines are first deleted. The remaining lines divide the plane into two regions, shaded white and purple. Two of the dashed lines can be restored, one crossing the inside region, the other crossing the outside. As there are no regions left, the remaining line cannot be restored without crossing either line. |Source={{own}} |Date= |Author= Cmglee |Permissi...
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3 utilities problem proof
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Proof without words that the three utilities problem has no solution by CMG Lee. The leftmost house is first deleted. The lines connecting the utilities with the remaining houses divide the plane into three regions, shaded yellow, red and blue. Whichever region the deleted house is placed into, one of the utilities is outside the region. The Jordan curve theorem states that a line connecting them must intersect one of the existing lines.
