Back in An Arrangement of 15 you were asked to place the numbers 1 to 15 in a line so that any two adjacent numbers summed to a square number.

Now, try to arrange the numbers from 1 to 32 in a circle, so any two adjacent numbers again sum a square number.

If we imagine a 32 points graph, such that there is an arc joining A and B if and only if A+B is a square, the problem is equivalent to finding a cycle in such a graph.

1 --> 3, 8, 15, 24


2 -->    7, 14, 23


3 --> 1, 6, 13, 22


4 -->    5, 12, 21, 32


5 -->    4, 11, 20, 31


6 -->    3, 10, 19, 30


7 -->    2,  9, 18, 29


8 -->    1,     17, 28


9 -->        7, 16, 27


10 ->        6, 15, 26


11 ->        5, 14, 25


12 ->        4, 13, 24


13 ->        3, 12, 23


14 ->        2, 11, 22


15 ->        1, 10, 21


16 ->            9, 20


17 ->            8, 19, 32


18 ->            7,     31


19 ->            6, 17, 30


20 ->            5, 16, 29


21 ->            4, 15, 28


22 ->            3, 14, 27


23 ->            2, 13, 26


24 ->            1, 12, 25


25 ->               11, 24


26 ->               10, 23


27 ->                9, 22


28 ->                8, 21


29 ->                7, 20


30 ->                6, 19


31 ->                5, 18


32 ->                4, 17

There's no simple algorithm for finding such a cycle, but we can simplify the graph by noting, for example, that from 28 we can only get to 8 and 21 -- we erase point 28, and remember that the graph will have 8-28-21. Doing this systematically arrives to the solution.

Posted by Old Original Oskar! on 2005-01-10 17:04:35