document.write( "Question 1203906: For some values of 𝑛, it is possible to cut a large square into 𝑛 smaller
\n" ); document.write( "squares (which need not be of equal size).
\n" ); document.write( "For example, it not possible when 𝑛 = 5 but is possible when 𝑛 = 6, as
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\n" ); document.write( "Find, with proof, all values of 𝑛 for which it is possible to do this. \n" ); document.write( "

Algebra.Com's Answer #839842 by mccravyedwin(407) $\"\"$ $\"About$

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document.write( "(1)  The first figure above proves that we can cut a square into 4 smaller\r\n" );
document.write( "squares.\r\n" );
document.write( "(2)  The second figure above proves that we can cut a square into 6 smaller\r\n" );
document.write( "squares.  \r\n" );
document.write( "(3)  The third figure above proves that we can cut a square into 8 smaller\r\n" );
document.write( "squares.\r\n" );
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document.write( "LEMMA:  If we can cut a square into n smaller squares,  we can also cut a square\r\n" );
document.write( "into n+3 smaller squares.\r\n" );
document.write( "Proof: Suppose we have a square cut into smaller squares. Now suppose we cut one\r\n" );
document.write( "of its smaller squares into 4 even smaller squares.  We then will have lost the\r\n" );
document.write( "square that we cut in the count, but we will have gained 4 squares, for a net\r\n" );
document.write( "gain of 3 squares.  QED\r\n" );
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document.write( "Therefore, by the LEMMA, we can also cut a square into 7 smaller squares, by\r\n" );
document.write( "picking one of the squares in the first figure and cutting it into 4 smaller\r\n" );
document.write( "squares.\r\n" );
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document.write( "So we have proved that we can cut a square into 6, 7, or 8 smaller squares.\r\n" );
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document.write( "Every integer divided by 3 leaves a remainder of 0, 1, or 2.\r\n" );
document.write( "So every integer is either a multiple of 3, 1 more than a multiple of 3,\r\n" );
document.write( "or 2 more than a multiple of 3.\r\n" );
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document.write( "Proof by induction: Assume that up through k, where , it is possible\r\n" );
document.write( "to cut a square into k smaller squares. Then by the LEMMA, a square can be cut\r\n" );
document.write( "into k+1 squares, by cutting a square in (k+1)-3 or k-2 smaller squares, and\r\n" );
document.write( "then cutting any one of its smaller squares into 4 even smaller squares, cutting\r\n" );
document.write( "the square into (k-2)+3 or k+1 squares.\r\n" );
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document.write( "So we can cut any square into n smaller squares except for n=2,3,5.  QED\r\n" );
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document.write( "Edwin

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