document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=190374>Question 190374</A>: <I><SPAN STYLE=\"word-break: break-all;\"> # 8.   Complete the magic square below (Three squares across, three down):\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #142877 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=142877\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> # 8  \r<BR>\n" );
document.write( "\n" );
document.write( "Q: <font color=red>Complete the magic square below (Three squares across, three down):</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's start with the given magic square:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/work/8-1.png\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The unique and defining property of any magic square is that the sum of the elements of any row, column, or diagonal will be the same.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So in this case, the sum of the elements in the 2nd row is: 11+7+3=21\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the sum of the elements in ANY row, column or diagonal will be 21. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's find the number in the lower right hand box. First, add up the elements in the 3rd column to get 8+3=11\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the sum must be 21, this means that we're off by 10 (since 21-11=10). So this means that 10 must go in the lower right hand corner. Let's add this number:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/work/8-2.png\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now let's find the element in the lower left hand corner. Add up the elements along the north-east diagonal (the diagonal from the lower left hand corner to the upper right hand corner) to get: 7+8=15\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since 21 - 15 = 6, this means that 6 must go in the lower left hand box. So let's add it in:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/work/8-3.png\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now let's find the number that belongs in the lower middle box. First, add the elements that we've found so far to get: 6 + 10 = 16\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Subtract this from 21 to get 21 - 16 = 5. So the missing number is 5 which means that 5 goes in the lower middle box like so:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/work/8-4.png\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now let's move up and find the element that goes in the upper left hand corner. First add the numbers in the first column to get: 11 + 6 = 17\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Subtract the result from 21 to obtain: 21 - 17 = 4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the number that belongs in the upper left hand box is 4. Add that in to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/work/8-5.png\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Finally, let's get the last number that belongs in the upper middle box. So add up the elements in the middle column to get: 7 + 5 = 12\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now take that result away from 21 to get: 21 - 12 = 9\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the number 9 belongs in the upper middle box. So our completed magic square looks like\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/work/8-6.png\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Check: let's add up the elements in each row, column and diagonal to verify the answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Row 1: 4 + 9 + 8 = 21 ... works\r<BR>\n" );
document.write( "\n" );
document.write( "Row 2: 11 + 7 + 3 = 21 ... works\r<BR>\n" );
document.write( "\n" );
document.write( "Row 3: 6 + 5 + 10 = 21 ... works\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Column 1: 4 + 11 + 6 = 21 ... works\r<BR>\n" );
document.write( "\n" );
document.write( "Column 2: 9 + 7 + 5 = 21 ... works\r<BR>\n" );
document.write( "\n" );
document.write( "Column 3: 8 + 3 + 10 = 21 ... works\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "North-West Diagonal: 4 + 7 + 10 = 21 ... works\r<BR>\n" );
document.write( "\n" );
document.write( "North-East Diagonal: 6 + 7 + 8 = 21 ... works\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since all of the sums are 21, this verifies the answer. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );