Algebra.Com's Answer #846843 by ikleyn(53299)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "When pipe is transported it is bundled into regular hexagons for stability during shipment.
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document.write( "Let n be the number of pieces of pipe on any side of the regular hexagon.
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document.write( "Write a rule for this situation. How many pieces of pipe are in a bundle when n = 12 ?
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document.write( "A regular hexagon consists of 6 congruent equilateral triangles.\r\n" );
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document.write( "Let's consider one such a triangle as a bundle of pipes.\r\n" );
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document.write( "The number of pipes in one such a triangled bundle, with n pipes along each side \r\n" );
document.write( "is the sum\r\n" );
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document.write( " 1 + 2 + 3 + . . . + n =  . (1)\r\n" );
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document.write( "It is the sum of first n natural numbers, so this formula is very well known.\r\n" );
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document.write( "OK. But we have 6 such triangles in the hexagon. So, the first move is to multiply \r\n" );
document.write( "the right side of (1) by 6 and to get 3n*(n+1).\r\n" );
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document.write( "But doing this way, we count n pipes twice along each common side of these triangles.\r\n" );
document.write( "So, from 3n(n+1) we should subtract 6n to get 3n*(n+1) - 6n.\r\n" );
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document.write( "But this is not the end.\r\n" );
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document.write( "When we multiplied (1) by 6, we counted the central pipe 6 times.\r\n" );
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document.write( "Then we subtracted it 6 times.\r\n" );
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document.write( "Now to compensate everything, we should add 1 for the central pipe.\r\n" );
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document.write( "So, the final formula is \r\n" );
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document.write( " f(n) = 3n*(n+1) - 6n + 1, (2)\r\n" );
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document.write( "or, which is the same\r\n" );
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document.write( " f(n) = 3n^2 - 3n + 1. (3)\r\n" );
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document.write( "You may check, using your pictures for small n = 2, 3, 4, that this formula is correct.\r\n" );
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document.write( " f(12) = using formula (2) = 3*12*13 - 6*12 + 1 = 397.\r\n" );
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document.write( "Solved, with explanations.\r
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document.write( "Thank you for asking.\r
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document.write( "Hope you will have fun reading this solution.\r
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