Let S and V represent the sum of all assigned numbers and the sum of the three numbers assigned to the edges forming any vertex, respectively.
Since the assigned numbers are distinct positive integers, S >= 78. But 8V = 2S --> S is a multiple of 4 so S >= 80.
Now S = 80 --> using the numbers {1,2,3,4,5,6,7,8,9,10,11,14} or {1,2,3,4,5,6,7,8,9,10,12,13} and V must equal 20.
By assigning the largest and smallest numbers in each set to edges forming the same vertex, and using some trial and error, the two following solutions are readily found:
AB=11,BC=2,CD=14,DA=1,EF=3,FG=10,GH=6,HE=9,AE=8,BF=7,CG=4,DH=5
AB=12,BC=5,CD=13,DA=1,EF=9,FG=8,GH=10,HE=4,AE=7,BF=3,CG=2,DH=6
Charlie's computer solution found six distinct assignments using these number sets. |
