Joe placed 10 numbered counters in a circle:

                  0

        1                   2




  8                               7





  4                               5




        6                   3

                  9
                  

He asked Penny to pick a number and then count clockwise from that numbered position on the circle that many further counters. For example, if she picked 7, the new counter would be 1 (located 7 clockwise from the 7). Repeating the process with the number on each new counter she would finish up at zero.

There was one particular number that, if she started there, she would go through all the other counters before getting to zero. What was that number?

Penny also found that, if she rearranged the counters in the circle in a certain order, leaving the topmost three (1, 0 and 2) in place (and maybe some others), that if she started with that same numbered counter from part 1, that she would also go through all the other counters before getting to zero. What was the clockwise order of the numbers in this new arrangement, starting with the zero at the top?

How would you create a magic circle of any value n? Is there an explicit form that gives magic circles for any n?

Posted by Gamer on 2006-12-17 00:41:36