document.write( "Question 1208226: Hi
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Algebra.Com's Answer #846512 by ikleyn(52814) $\"\"$ $\"About$

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\n" ); document.write( "A sheet of paper 42cm by 66cm has circles of 7cm radius cut out from it.
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\n" ); document.write( "\n" ); document.write( "        To solve it in the way as @MathLover1 does it is the same\r
\n" ); document.write( "\n" ); document.write( "        as to confess from the very beginning that you can not solve it\r
\n" ); document.write( "\n" ); document.write( "        in a right way and even do not know how to direct your thoughts in a right way.\r
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document.write( "To solve it in a right way, think about most dense packing congruent circles in a plane.  \r\n" );
document.write( "Such packing is placing the centers of circles in vertices of equilateral triangle grid on the plane.  \r\n" );
document.write( "The side length of these equilateral triangles is, obviously, the diameter of these circles 2*7 = 14 cm.\r\n" );
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document.write( "So, we place first row of circles on the line parallel to the long side of the\r\n" );
document.write( "rectangular piece of paper, placing their centers at the distance of 7 cm from the edge.\r\n" );
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document.write( "Four times the diameter is 4*14 = 56 cm, which is less than 66 cm; so we can place 4 circles this way.\r\n" );
document.write( "But five times the diameter is 5*14 = 70, which is greater than 66 cm, so 5-th circle does not fit.\r\n" );
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document.write( "In this page of paper, construct a grid of equilateral triangles with the side of 14 cm\r\n" );
document.write( "(which is the diameter of the circles).\r\n" );
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document.write( "So, one row of vertices is the centers of the first 4 circles, introduced above.\r\n" );
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document.write( "Next, second line of centers is the line remoted   =  12.12436 cm approximately (rounded UP)\r\n" );
document.write( "from the first line.\r\n" );
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document.write( "Let's estimate, how many circles can we place with the centers on the second line.\r\n" );
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document.write( "The center of the 1st circle will be 14 cm from the short edge, and the other centers will be placed\r\n" );
document.write( "on the second line with the step of 14 cm.\r\n" );
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document.write( "With 4 circles in the second row, we will have the horizontal extension in the second row\r\n" );
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document.write( "    14 + (14 + 14 + 14) + 7 = 63 cm,  which is less than 66 cm.\r\n" );
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document.write( "From it, we see that it is possible to place 4 and only 4 such circles with the centers \r\n" );
document.write( "on the second line.\r\n" );
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document.write( "Third line of the centers will be remoted   = 12.12436 cm from the second line.\r\n" );
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document.write( "Since  7 +  +  + 7 = 38.24872 cm  is less than 42 cm, \r\n" );
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document.write( "we see that the third row of circles fits vertically.\r\n" );
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document.write( "Thus, the conclusion from this reasoning is that 4 + 4 + 4 = 12 circles can be placed on / (or cut from)\r\n" );
document.write( "the given list of paper, and this is the MAXIMUM possible number of circles.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "Again, if somebody will show to a teacher or at a competition the solution similar to that by @MathLover1,
\n" ); document.write( "it can only make the teacher / (the jury) smile, since this way of thinking is inadequate to the problem.\r
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\n" ); document.write( "\n" ); document.write( "I am very glad to see that tutor @math_tutor2020 developed in his post even better solution
\n" ); document.write( "than mine. I am also very glad to see that the general idea of considering
\n" ); document.write( "the most dense packing works so well in this problem.\r
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