document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=774240>Question 774240</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve these magic squares, all rows,columns and diagonals all line up to the same number:<BR>\n" );
document.write( "-6     8     -2             ?        ?        5            ?       ?       ?<BR>\n" );
document.write( "?       ?        ?              ?      -12      ?            ?    -12    -16<BR>\n" );
document.write( " ?       -8     ?             ?        -8      3         -10     ?       ?\r<BR>\n" );
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document.write( "6       ?       4       -18<BR>\n" );
document.write( "?       ?       -6        ?<BR>\n" );
document.write( "?        ?      -8       10<BR>\n" );
document.write( "12     ?       ?        -12 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #472304 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20060)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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/cgi-bin/show-face.mpl?user=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=472304\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "I'll do only the last one, the hardest one.  The others \r\n" );
document.write( "are similar but easier. However there is no solution to \r\n" );
document.write( "the second one. Put different letters in place of the\r\n" );
document.write( " question marks:\r\n" );
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document.write( "A   B  5\r\n" );
document.write( "C -12  D\r\n" );
document.write( "E  -8  3\r\n" );
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document.write( "The diagonal sum 5-12+E (I'll label that Diag2) cannot equal to the\r\n" );
document.write( "bottom row sum (I'll label that Row3)\r\n" );
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document.write( "5-12+E=E-8+3\r\n" );
document.write( "  -7+E=E-5\r\n" );
document.write( "    -7=-5\r\n" );
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document.write( "A contradiction.  So a solution is impossible.\r\n" );
document.write( "Did you perchance type a number wrong?  (The others have solutions).   \r\n" );
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document.write( "Here's the last one, the hardest one:\r\n" );
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document.write( "Put different letters in place\r\n" );
document.write( "of all the question marks:\r\n" );
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document.write( " 6   A   4  -18\r\n" );
document.write( " B   C  -6    D\r\n" );
document.write( " E   F  -8   10\r\n" );
document.write( "12   G   H  -12\r\n" );
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document.write( "Write out the row, column, and diagonal sums\r\n" );
document.write( "and simplify them:\r\n" );
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document.write( " Row1: 6+A+4-18    = A-8             \r\n" );
document.write( " Row2: B+C-6+D     = B+C+D-6                    \r\n" );
document.write( " Row3: E+F-8+10    = E+F+2                   \r\n" );
document.write( " Row4: 12+G+H-12   = G+H                           \r\n" );
document.write( " Col1: 6+B+E+12    = B+E+18             \r\n" );
document.write( " Col2: A+C+F+G     = A+C+F+G              \r\n" );
document.write( " Col3: 4-6-8+H     = H-10                                       \r\n" );
document.write( " Col4: -18+D+10-12 = D-20            \r\n" );
document.write( " Diag1:0 6+C-8-12  = C-14                   \r\n" );
document.write( " Diag2: -18-6+F+12 = F-12                     \r\n" );
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document.write( "Row3 and Diag2 contain only E and F, so we set their sums equal:\r\n" );
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document.write( "E+F+2 = F-12\r\n" );
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document.write( "Subtract F from both sides:\r\n" );
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document.write( "    E = -14\r\n" );
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document.write( "Row4 and Col3 contain only G and H, so we set their sums equal:\r\n" );
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document.write( "G+H = H-10\r\n" );
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document.write( "Subtract H from both sides\r\n" );
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document.write( "   G = -10\r\n" );
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document.write( "So we substitute those in, and our list of sums get\r\n" );
document.write( "a couple more columns on the right\r\n" );
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document.write( "Row 1: 6+A+4-18    = A-8                 \r\n" );
document.write( "Row 2: B+C-6+D     = B+C+D-6                    \r\n" );
document.write( "Row 3: E+F-8+10    = E+F+2   = -14+F+2  = F-12                  \r\n" );
document.write( "Row 4: 12+G+H-12   = G+H     = -10+H    = H-10                      \r\n" );
document.write( "Col 1: 6+B+E+12    = B+E+18  = B-14+18  = B+4            \r\n" );
document.write( "Col 2: A+C+F+G     = A+C+F+G = A+C+F-10              \r\n" );
document.write( "Col 3: 4-6-8+H     = H-10                                       \r\n" );
document.write( "Col 4: -18+D+10-12 = D-20            \r\n" );
document.write( "Diag1: 6+C-8-12    = C-14                   \r\n" );
document.write( "Diag2: -18-6+F+12  = F-12\r\n" );
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document.write( "Now let's get things in terms of A:\r\n" );
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document.write( "A-8=F-12   A-8=H-10   A-8=B+4   A-8=D-20   A-8=C-14    \r\n" );
document.write( "A+4=F      A+2=H     A-12=B    A+12=D      A+6=C\r\n" );
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document.write( "Set Row1 = Row2, simplify, then substitute the above in the \r\n" );
document.write( "expression for Row2:    \r\n" );
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document.write( "              B+C+D-6=A-8\r\n" );
document.write( "              B+C+D+2=A  \r\n" );
document.write( "(A-12)+(A+6)+(A+12)+2=A\r\n" );
document.write( "      A-12+A+6+A+12+2=A\r\n" );
document.write( "                 3A+8=A\r\n" );
document.write( "                   2A=-8\r\n" );
document.write( "                    A=-4\r\n" );
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document.write( "So\r\n" );
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document.write( " A+4=F      A+2=H     A-12=B    A+12=D      A+6=C\r\n" );
document.write( "-4+4=F     -4+2=H    -4-12=B   -4+12=D     -4+6=C          \r\n" );
document.write( "   0=F       -2=H      -16=B       8=D        2=C\r\n" );
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document.write( "And we had found the values E=-14 and G=-10 first.\r\n" );
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document.write( "That's it.\r\n" );
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document.write( " 6   A   4  -18          6  -4   4  -18\r\n" );
document.write( " B   C  -6    D        -16   2  -6    8\r\n" );
document.write( " E   F  -8   10        -14   0  -8   10\r\n" );
document.write( "12   G   H  -12         12 -10  -2  -12\r\n" );
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document.write( "You do the others.\r\n" );
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document.write( "Edwin</PRE> </SPAN>\n" );
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