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Is it possible to draw 3 straight lines through a square such that they divide it into 7 equal-sized regions?

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  • 1
    $\begingroup$ By equal-sized do you mean of equal area? I mean, the shape doesn't matter. $\endgroup$ – Arnaud Mortier Sep 18 at 8:07
  • $\begingroup$ Assuming that we're focusing on area, ROT13(gurer ner 6 cnenzrgref naq 7 inevnoyrf, fhowrpg gb bar svkrq rdhngvba (gurve fhz vf gur nern bs gur fdhner) gurersber 6 cnenzrgref naq ernyyl 6 inevnoyrf, fb vg vf abg haernfbanoyr gb rkcrpg gung nyy pbzovangvbaf bs cnenzrgref pna or npuvrirq.) $\endgroup$ – Arnaud Mortier Sep 18 at 8:14
  • $\begingroup$ I mean of an Equal area in size $\endgroup$ – Dmitry Kamenetsky Sep 18 at 8:43
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Answer:

No, it's impossible.

More generally,

Buck and Buck, "Equipartition of Convex Sets", Mathematics Magazine 22(4) (1949), pp. 195-198, showed the following:

"We will be concerned here with three cuts; in general they divide a convex region into seven parts. If the cuts are concurrent, six parts result. We prove that division into six equal parts is always possible, but that although some regions can be divided into seven equal parts, no convex region can be so divided."

For the square,

since it is convex, it **cannot* be divided into seven equal parts by three straight lines.

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    $\begingroup$ It would be great (and a more self-contained answer) if you could add a rough idea of their proof. $\endgroup$ – Arnaud Mortier Sep 18 at 8:28

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