document.write( "Question 297613: How do I calculate all the possible rectangles that can fit into a 2D rectangle?
\n" ); document.write( "For instance if the large rectangle has height H = 2 and width W = 5 the total number of rectangles that can fit into it are 31. \r
\n" ); document.write( "\n" ); document.write( "what formula do I use to find all rectangles (including the large rectangle)that can fit into H x W? I tried dividing the rectangle by smaller values of H and W increasing, but still didn't get the right answer. Please help
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Algebra.Com's Answer #214319 by Fombitz(32388) $\"\"$ $\"About$

You can put this solution on YOUR website!
For a $\"H=2\"$ and $\"W=5\"$ rectangle, you get
\n" ); document.write( "10 $\"H=1\"$ , $\"W=1\"$ rectangles
\n" ); document.write( "8 $\"H=1\"$ , $\"W=2\"$ rectangles
\n" ); document.write( "6 $\"H=1\"$ , $\"W=3\"$ rectangles
\n" ); document.write( "4 $\"H=1\"$ , $\"W=4\"$ rectangles
\n" ); document.write( "2 $\"H=1\"$ , $\"W=5\"$ rectangles
\n" ); document.write( "5 $\"H=2\"$ , $\"W=1\"$ rectangles
\n" ); document.write( "4 $\"H=2\"$ , $\"W=2\"$ rectangles
\n" ); document.write( "3 $\"H=2\"$ , $\"W=3\"$ rectangles
\n" ); document.write( "2 $\"H=2\"$ , $\"W=4\"$ rectangles
\n" ); document.write( "1 $\"H=2\"$ , $\"W=5\"$ rectangles
\n" ); document.write( "That's a total of $\"45\"$ rectangles.
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\n" ); document.write( "Now for a formula.
\n" ); document.write( "Start simple and work up to see a pattern.
\n" ); document.write( " $\"H=2\"$ , $\"W=1\"$
\n" ); document.write( " $\"1x1=2\"$
\n" ); document.write( " $\"2x1=1\"$
\n" ); document.write( " $\"3\"$ total rectangles
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\n" ); document.write( " $\"+H=2\"$ , $\"+W=2\"$
\n" ); document.write( " $\"+1x1=+4\"$
\n" ); document.write( " $\"+1x2=+2\"$
\n" ); document.write( " $\"+2x1=+2\"$
\n" ); document.write( " $\"+2x2=+1\"$
\n" ); document.write( " $\"+9+\"$ total rectangles.
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\n" ); document.write( " $\"+H=2\"$ , $\"+W=3\"$
\n" ); document.write( " $\"+1x1=6\"$
\n" ); document.write( " $\"+1x2=4\"$
\n" ); document.write( " $\"+1x3=2\"$
\n" ); document.write( " $\"+2x1=3\"$
\n" ); document.write( " $\"+2x2=2\"$
\n" ); document.write( " $\"+2x3=1\"$
\n" ); document.write( " $\"+18+\"$ total
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\n" ); document.write( " $\"+H=2\"$ , $\"+W=4\"$
\n" ); document.write( " $\"+1x1=8\"$
\n" ); document.write( " $\"+1x2=6\"$
\n" ); document.write( " $\"+1x3=4\"$
\n" ); document.write( " $\"+1x4=2\"$
\n" ); document.write( " $\"+2x1=4\"$
\n" ); document.write( " $\"+2x2=3\"$
\n" ); document.write( " $\"+2x3=2\"$
\n" ); document.write( " $\"+2x4=1\"$
\n" ); document.write( " $\"+30\"$ total rectangles.
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\n" ); document.write( ".
\n" ); document.write( "You're starting to get a pattern.
\n" ); document.write( "The $\"2xn\"$ rectangles sum from $\"1\"$ to $\"n\"$ in steps of $\"1\"$ for a total of $\"n%28n%2B1%29%2F2\"$ .
\n" ); document.write( "The $\"1xn\"$ rectangles sum for $\"2\"$ to $\"2n\"$ in steps of $\"2\"$ or $\"n%28n%2B1%29\"$ .
\n" ); document.write( "So the grand total of rectangles in a $\"2xn\"$ rectangle would be
\n" ); document.write( " $\"S%28n%29=n%28n%2B1%29%2F2%2Bn%28n%2B1%29\"$
\n" ); document.write( " $\"S%28n%29=%283n%2F2%29%28n%2B1%29\"$ \n" ); document.write( "

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