document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1040063>Question 1040063</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many ways are there to re-arrange numbers 1 to 12 on a circle, so that the sum of any 3 consecutive numbers is divisible by 3? <BR>\n" );
document.write( "Obviously in the original arrangement (1, 2, 3, ..., 12) this is true because the sum of any 3 consecutive integers is always divisible by 3 (n, n+1, n+2 ---> 3n+3). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #654869 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=654869\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> When we are talking about multiples of 3,<BR>\n" );
document.write( "all numbers are \"congruent modulo 3\" with either 1, 2 or 0.<BR>\n" );
document.write( "We could say \"congruent modulo 3 to 1, 2, or 3\" instead),<BR>\n" );
document.write( "There are only two ways to arrange the numbers<BR>\n" );
document.write( "1, 2, and 3 (each repeated 4 times) around a circle,<BR>\n" );
document.write( "so that the sum of any 3 consecutive numbers would be a multiple of 3,<BR>\n" );
document.write( "and one way is the mirror image of the other:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C250%2C-2.5%2C2.5%2C-1.25%2C1.25%2C%0D%0Acircle%28-.15%2C0%2C0.1%29%2Ccircle%28-0.28%2C0.5%2C0.1%29%2C%0D%0Acircle%28-.65%2C0.87%2C0.1%29%2Ccircle%28-1.15%2C1%2C0.1%29%2C%0D%0Acircle%28-1.65%2C0.87%2C0.1%29%2Ccircle%28-2.02%2C-0.5%2C0.1%29%2C%0D%0Acircle%28-2.15%2C0%2C0.1%29%2Ccircle%28-2.02%2C0.5%2C0.1%29%2C%0D%0Acircle%28-1.65%2C-0.87%2C0.1%29%2Ccircle%28-1.15%2C-1%2C0.1%29%2C%0D%0Acircle%28-0.65%2C-0.87%2C0.1%29%2Ccircle%28-0.28%2C-0.5%2C0.1%29%2C%0D%0Acircle%282.15%2C0.00%2C0.1%29%2Ccircle%282.02%2C0.50%2C0.1%29%2C%0D%0Acircle%281.65%2C0.87%2C0.1%29%2Ccircle%281.15%2C1.00%2C0.1%29%2C%0D%0Acircle%280.65%2C0.87%2C0.1%29%2Ccircle%280.28%2C0.50%2C0.1%29%2C%0D%0Acircle%280.15%2C0.00%2C0.1%29%2Ccircle%280.28%2C-0.50%2C0.1%29%2C%0D%0Acircle%280.65%2C-0.87%2C0.1%29%2Ccircle%281.15%2C-1.00%2C0.1%29%2C%0D%0Acircle%281.65%2C-0.87%2C0.1%29%2Ccircle%282.02%2C-0.50%2C0.1%29%2C%0D%0Alocate%28-0.18%2C0.08%2C1%29%2Clocate%28-0.31%2C0.58%2C2%29%2Clocate%28-0.68%2C0.95%2C3%29%2C%0D%0Alocate%28-1.18%2C1.08%2C1%29%2Clocate%28-1.68%2C0.95%2C2%29%2Clocate%28-2.05%2C0.58%2C3%29%2C%0D%0Alocate%28-2.18%2C0.08%2C1%29%2Clocate%28-2.05%2C-0.42%2C2%29%2Clocate%28-1.68%2C-0.79%2C3%29%2C%0D%0Alocate%28-1.18%2C-0.92%2C1%29%2Clocate%28-0.68%2C-0.79%2C2%29%2Clocate%28-0.31%2C-0.42%2C3%29%2C%0D%0Alocate%280.12%2C0.08%2C1%29%2Clocate%280.25%2C0.58%2C2%29%2Clocate%280.62%2C0.95%2C3%29%2C%0D%0Alocate%281.12%2C1.08%2C1%29%2Clocate%281.62%2C0.95%2C2%29%2Clocate%281.99%2C0.58%2C3%29%2C%0D%0Alocate%282.12%2C0.08%2C1%29%2Clocate%281.99%2C-0.42%2C2%29%2Clocate%281.62%2C-0.79%2C3%29%2C%0D%0Alocate%281.12%2C-0.92%2C1%29%2Clocate%280.62%2C-0.79%2C2%29%2Clocate%280.25%2C-0.42%2C3%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Turning the circle around may make the arrangement look different, but it is the same arrangement.<BR>\n" );
document.write( "If we read the numbers in sequence starting from 1,<BR>\n" );
document.write( "we have 1,2,3,1,2,3,1,2,3,1,2,3 clockwise or counterclockwise.<BR>\n" );
document.write( "To use the numbers from 1 to 12, we just replace each number with a different congruent number.<BR>\n" );
document.write( "Listing 1 as the first number,<BR>\n" );
document.write( "the fourth number could be 4, 7, or 10 (all congruent to 1 modulo 3);<BR>\n" );
document.write( "the second, fifth, eighth, and eleventh number must be 2, 5, 8, and 11, in some order,<BR>\n" );
document.write( "and the third, sixth, ninth and twelfth numbers must be 3, 6, 9, and 12 in some order.<BR>\n" );
document.write( "Staring our list with number 1, there are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%21=6 ALIGN=MIDDLE  ALT=\"3%21=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible arrangements for 4, 7, and 10,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%21=24 ALIGN=MIDDLE  ALT=\"4%21=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible arrangements for 2, 5, 8, and 11,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%21=24 ALIGN=MIDDLE  ALT=\"4%21=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible arrangements for 3, 6, 9, and 12.<BR>\n" );
document.write( "All in all we could make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2A24%2A24=3456 ALIGN=MIDDLE  ALT=\"6%2A24%2A24=3456\"   DISABLED_style= \"max-width:90%; height:auto;\" > different lists,<BR>\n" );
document.write( "that can be arrange in two directions around the circle to make =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A3456=highlight%286912%29 ALIGN=MIDDLE  ALT=\"2%2A3456=highlight%286912%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "ways to arrange numbers 1 to 12 on a circle, so that the sum of any 3 consecutive numbers is divisible by 3.\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );