Number Puzzle #6 – Solution
February 6th, 2009 by SplineGuy
Two days ago, I posted this simple little number puzzle. Quite a few folks came up with the answer below. One of the interesting questions you can ask is whether that solution is unique.
Clearly there are two lines of symmetry in the original problem so by reflection alone we come up with a total of four solutions: 
where 
and 
represent “flips” across the lines of symmetry and 
represents the identity, or the solution above. By 
, I mean the composition of the flipping operations or just consecutive flipping.
There also exists radial symmetry at 
, but this is equivalent to . So for this arrangement above, there are four solutions of the same “type”.
Are any other arrangements possible besides these four?
1 and 8 must go on the central 2 points, as they are the only values with 6 non-neighbours.
2 must go top/bottom away from 1 as all other remaining nodes are connected to 1, and the same works for 7 (the numbers are symmetric about 4|5).
3 and 6 now similarly have only one position available each, leaving 4 and 5.
So the answer is no, there are no further arrangements.