Arrange the numbers from 1 to 15 in such an order that any two consecutive numbers in the sequence add up to a perfect square.

8, 1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9
or its reflection
9, 7, 2, 14, 11, 5, 4, 12, 13, 3, 6, 10, 15, 1, 8

The perfect squares greater than the sum of the two smallest numbers and less than the sum of the two largest numbers are 4, 9, 16 and 25. Each number, except those that begin or end the sequence must be paired to create two squares, one to its fore and one to its aft, in the sequence. Due to this requirement, every other pair will total 16 with alternating pairs, fore and aft, totaling 9 and 25:

8,1                                 ( 9) 
  1,15                              (16) 
    15,10                           (25)
       10,6                         (16)
          6,3                       ( 9)
            3,13                    (16)
              13,12                 (25)
                 12,4               (16)
                    4,5             ( 9)
                      5,11          (16)
                        11,14       (25)
                           14,2     (16)
                              2,7   ( 9)
                                7,9 (16)
 

Posted by Dej Mar on 2010-07-04 03:59:54