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I have just read that a square can't be divided into 2 , 3 or 5 smaller squares ( not necessarily distinct). I am unable to think of a way to prove it. can anyone help me by providing a hint?

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  • $\begingroup$ What does "not necessarily distinct" mean? Distinct sizes? $\endgroup$ – Travis Mar 8 '16 at 12:18
  • $\begingroup$ Smaller squares may or may not have same area. $\endgroup$ – rugi Mar 8 '16 at 12:19
  • $\begingroup$ I have found a related thread math.stackexchange.com/questions/1337337/… $\endgroup$ – rugi Mar 8 '16 at 12:24
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If $n<4$ , at least one of the squares must cover two vertices of the original square. These vertices must also be vertices of the smaller square and it must not protrude; we conclude that it has side length $\ge 1$ and covers the big square by itself

For $n=5$, ther must be exactly one small square that does not cover a vertex of the original square. Only few configurations of neighbourship are possible, and none of the works.

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  • $\begingroup$ Aha , extremal principle in action. Thanks for the wonderful proof. $\endgroup$ – rugi Mar 8 '16 at 12:31
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When a square gets divided in to more squares there are (divisions)*(divisions) new squares. If you divide a square two times you get 4 squares.

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  • $\begingroup$ I have just arrived at this conclusion for n=2. But what about 3 and 5? $\endgroup$ – rugi Mar 8 '16 at 12:10
  • $\begingroup$ why is this true for square and not triangle? $\endgroup$ – Arjang Mar 8 '16 at 12:13
  • $\begingroup$ @Arjang for triangle one can draw a perpendicular to opposite side and divide it into two different triangle. $\endgroup$ – rugi Mar 8 '16 at 12:17
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    $\begingroup$ @Arjihad three of side length $\frac 13$ on top, two more on the right, another of side length $\frac23$ for the rest $\endgroup$ – Hagen von Eitzen Mar 8 '16 at 12:24
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    $\begingroup$ @GerryMyerson Except n=2 , 3 and 5 a square can be sub divided into other squares for any other n. See a related thread math.stackexchange.com/questions/1337337/… $\endgroup$ – rugi Mar 8 '16 at 12:25

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