Can you assign twelve distinct positive integers, one to each edge of a cube, so that the sum of the three numbers assigned to the edges forming a vertex is always the same, and the total sum of the assigned numbers is minimized?

For referencing purposes, let the cube have vertices with coordinates: A(0,0,0), B(1,0,0), C(1,1,0), D(0,1,0), E(0,0,1), F(1,0,1), G(1,1,1), and H(0,1,1), so that the edges are: AB, BC, CD, DA, EF, FG, GH, HE, AE, BF, CG, and DH.

As the integers are all different and positive, they must add up to at least the sum of the first 12 positive integers, or 78.

Each edge value counts into two vertices, so the total of the vertices' values is twice the total of the edge values. The average, then, of the eight vertices is then twice the edge total divided by eight. As each vertex is to have the same total value, that value must be the average value, or the total of the edges, divided by four.

Thus the total must be divisible by four. The first number divisible by four and at least 78 is 80.  The following program starts at 80, and in fact need not have looked further, as there are solutions where the total is 80.

Dim goAhead, getOut
Private Sub Command1_Click()
 MousePointer = vbHourglass
 For total = 80 To 200 Step 4
  DoEvents
  Print total
  eachVertex = total / 4
  ReDim Number(12)
  For n1 = 1 To (total - 1) / 12: Number(1) = n1: sofar1 = Number(1): DoEvents
  For n2 = n1 + 1 To (total - sofar1 - 1) / 11: Number(2) = n2: sofar2 = sofar1 + Number(2): DoEvents
  For n3 = Number(2) + 1 To (total - sofar2 - 1) / 10: Number(3) = n3: sofar3 = sofar2 + Number(3): DoEvents
  For n4 = Number(3) + 1 To (total - sofar3 - 1) / 9: Number(4) = n4: sofar4 = sofar3 + Number(4): DoEvents
  For n5 = Number(4) + 1 To (total - sofar4 - 1) / 8: Number(5) = n5: sofar5 = sofar4 + Number(5): DoEvents
  For n6 = Number(5) + 1 To (total - sofar5 - 1) / 7: Number(6) = n6: sofar6 = sofar5 + Number(6): DoEvents
  For n7 = Number(6) + 1 To (total - sofar6 - 1) / 6: Number(7) = n7: sofar7 = sofar6 + Number(7): DoEvents
  For n8 = Number(7) + 1 To (total - sofar7 - 1) / 5: Number(8) = n8: sofar8 = sofar7 + Number(8): DoEvents
  For n9 = Number(8) + 1 To (total - sofar8 - 1) / 4: Number(9) = n9: sofar9 = sofar8 + Number(9): DoEvents
  For n10 = Number(9) + 1 To (total - sofar9 - 1) / 3: Number(10) = n10: sofar10 = sofar9 + Number(10): DoEvents
  For n11 = Number(10) + 1 To (total - sofar10 - 1) / 2: Number(11) = n11: sofar11 = sofar10 + Number(11): DoEvents
  For n12 = Number(11) + 1 To (total - sofar11): Number(12) = n12: sofar12 = sofar11 + Number(12): DoEvents
   If sofar12 = total Then
    ReDim edge(12), used(12)
    edge(1) = Number(1)
    used(1) = 1
    For e2sub = 2 To 11
     edge(2) = Number(e2sub)
     used(e2sub) = 1
    For e3sub = e2sub + 1 To 12
     edge(3) = Number(e3sub)
     used(e3sub) = 1
     If edge(1) + edge(2) + edge(3) = eachVertex Then
    
      For e4sub = 2 To 12
       If used(e4sub) = 0 Then
        used(e4sub) = 1
        edge(4) = Number(e4sub)
        For e5sub = 2 To 12
         If used(e5sub) = 0 Then
          used(e5sub) = 1
          edge(5) = Number(e5sub)
          If edge(1) + edge(4) + edge(5) = eachVertex Then
         
      For e6sub = 2 To 12
       If used(e6sub) = 0 Then
        used(e6sub) = 1
        edge(6) = Number(e6sub)
        For e7sub = 2 To 12
         If used(e7sub) = 0 Then
          used(e7sub) = 1
          edge(7) = Number(e7sub)
          If edge(4) + edge(6) + edge(7) = eachVertex Then
         
      For e8sub = 2 To 12
       If used(e8sub) = 0 Then
        used(e8sub) = 1
        edge(8) = Number(e8sub)
          If edge(2) + edge(6) + edge(8) = eachVertex Then
         
      For e9sub = 2 To 12
       If used(e9sub) = 0 Then
        used(e9sub) = 1
        edge(9) = Number(e9sub)
        For e10sub = 2 To 12
         If used(e10sub) = 0 Then
          used(e10sub) = 1
          edge(10) = Number(e10sub)
          If edge(8) + edge(9) + edge(10) = eachVertex Then
         
      For e11sub = 2 To 12
       If used(e11sub) = 0 Then
        used(e11sub) = 1
        edge(11) = Number(e11sub)
          If edge(11) + edge(3) + edge(10) = eachVertex Then
            For subscr = 1 To 12
             If used(subscr) = 0 Then
              edge(12) = Number(subscr)
              If edge(12) + edge(5) + edge(11) = eachVertex And edge(12) + edge(7) + edge(9) = eachVertex Then
               For i = 1 To 12
                Print edge(i);
               Next
               Print
              End If
             End If
            Next
            Open "c:\progra~1\devstudio\vb\projects\flooble\edge-numbered cube.txt" For Append As #2
             Print #2, "GH CG  FG  DH HE CD DA  BC AB BF  EF AE"
               For i = 1 To 12
                Print #2, edge(i);
               Next
               Print #2, "    "; total
             Print #2, "    D+-"; Right("0" + LTrim(Str(edge(6))), 2); "-+C"
             Print #2, "     |    |"
             Print #2, "    "; Right("0" + LTrim(Str(edge(4))), 2); "    "; Right("0" + LTrim(Str(edge(2))), 2)
             Print #2, "D   H|    |G   C"
             Print #2, "+-"; Right("0" + LTrim(Str(edge(4))), 2); "-+-"; Right("0" + LTrim(Str(edge(1))), 2); "-+-"; Right("0" + LTrim(Str(edge(2))), 2); "-+"
             Print #2, "|    |    |    |"
             Print #2, Right("0" + LTrim(Str(edge(7))), 2); "   "; Right("0" + LTrim(Str(edge(5))), 2); "   "; Right("0" + LTrim(Str(edge(3))), 2); "   "; Right("0" + LTrim(Str(edge(8))), 2)
             Print #2, "|A  E|    |F  B|"
             Print #2, "+-"; Right("0" + LTrim(Str(edge(12))), 2); "-+-"; Right("0" + LTrim(Str(edge(11))), 2); "-+-"; Right("0" + LTrim(Str(edge(10))), 2); "-+"
             Print #2, "     |    |"
             Print #2, "    "; Right("0" + LTrim(Str(edge(12))), 2); "    "; Right("0" + LTrim(Str(edge(10))), 2)
             Print #2, "     |    |"
             Print #2, "    A+-"; Right("0" + LTrim(Str(edge(9))), 2); "-+B"
             Print #2,
            Close 2
            goAhead = False
            MousePointer = vbDefault
            Do: DoEvents: Loop Until goAhead = True Or getOut = True
            If getOut Then Exit Sub
            MousePointer = vbHourglass
          End If
        used(e11sub) = 0
       End If
      Next e11sub
         
          End If
          used(e10sub) = 0
         End If
        Next e10sub
        used(e9sub) = 0
       End If
      Next e9sub
         
          End If
        used(e8sub) = 0
       End If
      Next e8sub

          End If
          used(e7sub) = 0
         End If
        Next e7sub
        used(e6sub) = 0
       End If
      Next e6sub
         
          End If
          used(e5sub) = 0
         End If
        Next e5sub
        used(e4sub) = 0
       End If
      Next e4sub
     
     End If
     used(e3sub) = 0
    Next e3sub
     used(e2sub) = 0
    Next e2sub
    used(1) = 0
   End If
  Next n12
  Next n11
  Next n10
  Next n9
  Next n8
  Next n7
  Next n6
  Next n5
  Next n4
  Next n3
  Next n2
  Next n1

 Next
 MousePointer = vbDefault
End Sub

Private Sub Command2_Click()
  goAhead = True
End Sub

Private Sub Command3_Click()
  getOut = True
End Sub

The program always places the lowest number in the top front edge, which is GH according to the puzzle statement and understanding the z-axis to be positive toward you.

The program also chooses CG to be smaller than FG, but that's the limit of the degree toward which trivial rotations and reflections are weeded out by the program. I weeded out other rotations/reflections by editing the output, leaving only those solutions where the lowest valued edge adjacent to GH is CG. There are six left, that are therefore non-trivial variations, with this lowest total, 80, with each vertex adding to 20.

Note in the diagrams that CG, DH, AE and BF are shown twice, as the faces they separate are split from one another.

GH CG  FG  DH HE CD DA  BC AB BF  EF AE
 1  5  14  8  11  9  3  6  10  4  2  7      80 
 
    D+-09-+C
     |    |
    08    05
D   H|    |G   C
+-08-+-01-+-05-+
|    |    |    |
03   11   14   06
|A  E|    |F  B|
+-07-+-02-+-04-+
     |    |
    07    04
     |    |
    A+-10-+B

GH CG  FG  DH HE CD DA  BC AB BF  EF AE
 1  5  14  9  10  8  3  7  11  2  4  6      80 

    D+-08-+C
     |    |
    09    05
D   H|    |G   C
+-09-+-01-+-05-+
|    |    |    |
03   10   14   07
|A  E|    |F  B|
+-06-+-04-+-02-+
     |    |
    06    02
     |    |
    A+-11-+B

GH CG  FG  DH HE CD DA  BC AB BF EF AE
 1  5  14  11  8  6  3  9  7  4  2  10      80 

    D+-06-+C
     |    |
    11    05
D   H|    |G   C
+-11-+-01-+-05-+
|    |    |    |
03   08   14   09
|A  E|    |F  B|
+-10-+-02-+-04-+
     |    |
    10    04
     |    |
    A+-07-+B

GH CG  FG  DH HE CD DA  BC AB BF  EF AE
 1  6  13  7  12  4  9  10  8  2  5  3      80 

    D+-04-+C
     |    |
    07    06
D   H|    |G   C
+-07-+-01-+-06-+
|    |    |    |
09   12   13   10
|A  E|    |F  B|
+-03-+-05-+-02-+
     |    |
    03    02
     |    |
    A+-08-+B

GH CG  FG  DH HE CD DA  BC AB BF  EF AE
 1  6  13  10  9  2  8  12  5  3  4  7      80 

    D+-02-+C
     |    |
    10    06
D   H|    |G   C
+-10-+-01-+-06-+
|    |    |    |
08   09   13   12
|A  E|    |F  B|
+-07-+-04-+-03-+
     |    |
    07    03
     |    |
    A+-05-+B

GH CG  FG  DH HE CD DA  BC AB BF  EF AE
 1  7  12  9  10  8  3  5  13  2  6  4      80 

    D+-08-+C
     |    |
    09    07
D   H|    |G   C
+-09-+-01-+-07-+
|    |    |    |
03   10   12   05
|A  E|    |F  B|
+-04-+-06-+-02-+
     |    |
    04    02
     |    |
    A+-13-+B

 

Posted by Charlie on 2008-03-19 18:41:48