The network represents a map of the edges and vertices of a
hypercube or more precisely, a tesseract when imposed within 2D space.
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The orange squares represent the faces of the internal cell, a cube in 3D space, and is defined by vertices A to H. The larger bolder letters, I to P are the vertices of the outer cell/cube.
The diagonal lines are the edges which link the inner to the outer forming the faces that configure the cells which interface to the inner and outer cells.
Let each hypercell have a value which is the vertex sum of each of its enclosing faces (or three times the sum of its 8 vertices).
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The vertices are to be numbered uniquely from 1 to 16, and for consistency of reference, let
A always be 1.
Find sets of values for A through P so that all the cells have the same value.
Note: while the on-line calculator may be useful, a spreadsheet should prove more valuable for those able to use one.
The fact that the total of the vertices of each cell is 68 enables pruning the tree before all permutations of 1 - 16 are tried.
Dim used(16)
Private Sub Form_Load()
Me.Visible = True
Open "hypercell.txt" For Output As #2
txtOut.Text = " b c d e f g h i j k l m n o p"
Print #2, txtOut.Text
a = 1
For b = 2 To 13
used(b) = 1
For c = b + 1 To 14
If used(c) = 0 Then
used(c) = 1
For i = c + 1 To 15
If used(i) = 0 Then
used(i) = 1
CurrentX = 1: CurrentY = 1
Print b; c; i; " "; ct; " "
For f = i + 1 To 16
If used(f) = 0 Then
used(f) = 1
For e = 2 To 16
If used(e) = 0 Then
used(e) = 1
ae = a + b + c + f + e
If ae <= 59 Then
For d = 2 To 16
If used(d) = 0 Then
used(d) = 1
af = ae + d
If af <= 63 Then
For g = 2 To 16
If used(g) = 0 Then
used(g) = 1
ag = af + g
If ag <= 66 Then
For h = 2 To 16
If used(h) = 0 Then
used(h) = 1
If a + b + c + d + e + f + g + h = 68 Then ' C 1
DoEvents
abefi = a + b + e + f + i
If abefi <= 59 Then
For j = 2 To 16
If used(j) = 0 Then
used(j) = 1
abefij = abefi + j
If abefij <= 63 Then
For m = 2 To 16
If used(m) = 0 Then
used(m) = 1
abefijm = abefij + m
If abefijm <= 66 Then
For n = 2 To 16
If used(n) = 0 Then
used(n) = 1
abefijmn = abefijm + n
If abefijmn = 68 Then ' C 4
abcdij = a + b + c + d + i + j
kl = 68 - abcdij ' C 5
For k = 1 To 16
If used(k) = 0 Then
used(k) = 1
l = kl - k
If l <= 16 And l > 1 Then
If used(l) = 0 Then
used(l) = 1
p = 68 - a - d - f - g - i - l - m ' C 3
If p <= 16 And p > 1 Then
If used(p) = 0 Then
used(p) = 1
o = 68 - b - c - e - h - j - k - n ' C 6
If o <= 16 And o > 1 Then
If used(o) = 0 Then
used(o) = 1
If i + j + k + l + m + n + o + p = 68 Then
If c + d + g + h + k + l + o + p = 68 Then
If e + f + g + h + m + n + o + p = 68 Then
textOut = ""
textOut = textOut + Right(" " + Format(b, " ##"), 3)
textOut = textOut + Right(" " + Format(c, " ##"), 3)
textOut = textOut + Right(" " + Format(d, " ##"), 3)
textOut = textOut + Right(" " + Format(e, " ##"), 3)
textOut = textOut + Right(" " + Format(f, " ##"), 3)
textOut = textOut + Right(" " + Format(g, " ##"), 3)
textOut = textOut + Right(" " + Format(h, " ##"), 3)
textOut = textOut + Right(" " + Format(i, " ##"), 3)
textOut = textOut + Right(" " + Format(j, " ##"), 3)
textOut = textOut + Right(" " + Format(k, " ##"), 3)
textOut = textOut + Right(" " + Format(l, " ##"), 3)
textOut = textOut + Right(" " + Format(m, " ##"), 3)
textOut = textOut + Right(" " + Format(n, " ##"), 3)
textOut = textOut + Right(" " + Format(o, " ##"), 3)
textOut = textOut + Right(" " + Format(p, " ##"), 3)
Print #2, textOut
txtOut.Text = txtOut.Text & Chr$(13) & Chr$(10) & textOut
ct = ct + 1
End If
End If
End If
used(o) = 0
End If
End If ' o <=16
used(p) = 0
End If
End If ' p <= 16
used(l) = 0
End If 'unused l
End If ' l <=16
used(k) = 0
End If
Next k
End If 'mabefijmn = 68
used(n) = 0
End If
Next n
End If 'mabefijm <= 66
used(m) = 0
End If
Next m
End If 'abefij <= 63
used(j) = 0
End If
Next j
End If ' abefi <= 59
End If ' a thru h = 68
used(h) = 0
End If
Next h
End If 'ag<=52
used(g) = 0
End If
Next g
End If ' af<=37
used(d) = 0
End If
Next d
End If ' ae <=59
used(e) = 0
End If
Next e
used(f) = 0
End If
Next f
used(i) = 0
End If
Next i
used(c) = 0
End If
Next c
used(b) = 0
Next b
Print ct
Close 2
End Sub
This program finds 162,276 solutions.
The program assumes that A is 1 and that B < C < I < F, so as to find only one out of any rotations and reflections.
Reflections turn this into 324,552.
You might want to multiply that by 12 for rotations; then you can also multiply by 16 if you allow for A to be other than 1. But I think those are overkill; there are really only 324,552 that are fundamentally different in 3-space or 162,276 solutions that are fundamentally different mathematically or topologically. This still seems like a lot of solutions, but consider that the number of overall ways of arranging the numbers 1 - 16 that are topologically different is 16!/(6*16) = 217,945,728,000.
Looked at another way, the rotations and reflections had been eliminated by requiring B < C < I < F, only 1/24 of the possibilities.
A random sampling of the 162,276 solutions is shown below. Each solution was given a 1/1000 probability of being shown (if rnd(1) < 1/1000). As it turns out, 165 were selected.
In all solutions, a = 1.
b c d e f g h i j k l m n o p
2 4 14 16 12 9 10 11 15 8 13 5 6 7 3
2 7 16 13 15 4 10 11 14 5 12 3 9 8 6
2 7 14 15 16 4 9 11 12 8 13 6 5 10 3
2 7 16 11 14 5 12 13 15 4 10 3 9 8 6
2 8 14 13 16 5 9 10 12 6 15 3 11 7 4
2 8 15 16 13 3 10 12 14 5 11 4 6 7 9
2 8 11 7 16 14 9 12 15 13 6 5 10 4 3
2 9 16 7 15 6 12 14 10 13 3 8 11 4 5
2 10 13 15 12 8 7 11 9 16 6 14 4 5 3
2 10 12 8 16 5 14 13 15 4 11 7 6 9 3
2 12 10 8 16 4 15 13 14 9 7 11 3 5 6
2 12 11 7 16 13 6 15 14 9 4 3 10 8 5
2 13 9 10 16 12 5 14 11 15 3 6 8 4 7
2 14 5 13 16 10 7 15 11 12 8 4 6 3 9
3 4 14 15 12 6 13 9 16 10 11 7 5 2 8
3 4 14 11 16 6 13 9 15 10 12 8 5 7 2
3 5 15 13 12 11 8 10 16 14 4 6 7 2 9
3 6 14 15 12 4 13 9 16 8 11 10 2 5 7
3 6 16 14 12 9 7 10 11 8 13 2 15 4 5
3 6 13 8 14 7 16 10 15 11 9 12 5 4 2
3 6 15 9 14 4 16 12 11 13 7 10 8 2 5
3 7 12 16 15 5 9 13 10 8 14 6 4 11 2
3 7 13 11 16 8 9 15 14 10 5 6 2 12 4
3 8 14 13 15 2 12 10 16 7 9 6 4 5 11
3 8 12 14 15 5 10 11 16 4 13 2 6 7 9
3 8 12 11 16 2 15 14 10 7 13 4 9 5 6
3 9 16 8 15 2 14 10 13 4 12 7 11 6 5
3 9 12 14 16 6 7 15 4 13 11 5 10 8 2
3 11 15 12 16 2 8 14 5 13 6 10 7 9 4
3 12 10 4 14 9 15 13 16 5 8 11 6 7 2
3 12 6 8 16 7 15 13 14 10 9 11 2 4 5
3 14 13 10 16 7 4 15 5 9 8 6 12 11 2
4 5 10 14 15 12 7 8 11 16 13 6 9 2 3
4 5 10 8 15 14 11 12 16 13 7 3 9 2 6
4 5 12 14 15 8 9 13 11 16 6 3 7 2 10
4 5 8 16 15 10 9 13 14 12 11 3 2 6 7
4 5 8 10 16 11 13 15 14 9 12 2 6 7 3
4 6 16 15 11 10 5 7 8 14 12 9 13 3 2
4 6 15 12 9 10 11 8 13 14 7 16 5 3 2
4 6 7 10 15 12 13 9 16 11 14 8 5 3 2
4 6 14 15 12 3 13 10 8 9 16 7 11 2 5
4 6 8 12 14 16 7 10 13 15 11 5 9 2 3
4 7 15 11 13 12 5 9 16 14 2 6 8 3 10
4 7 15 16 13 3 9 10 11 14 6 8 5 2 12
4 7 14 15 16 9 2 11 10 13 8 6 5 12 3
4 7 9 13 15 3 16 14 10 11 12 6 5 2 8
4 7 10 13 16 11 6 15 14 9 8 2 3 12 5
4 8 14 16 13 10 2 9 15 11 6 3 7 5 12
4 8 13 14 12 11 5 10 9 7 16 3 15 6 2
4 8 10 12 16 2 15 13 7 11 14 9 6 5 3
4 8 11 9 16 6 13 15 14 10 5 2 7 3 12
4 9 16 7 15 2 14 12 13 10 3 11 5 6 8
4 9 15 11 16 2 10 12 6 7 14 5 13 8 3
4 10 15 5 14 8 11 12 13 7 6 3 16 2 9
4 10 16 14 15 5 3 12 6 11 8 9 7 13 2
4 10 7 15 14 9 8 13 12 5 16 6 3 11 2
4 10 13 16 15 2 7 14 3 11 12 6 9 8 5
4 11 10 9 14 16 3 12 13 15 2 8 7 6 5
4 11 14 16 15 2 5 12 9 7 10 8 3 13 6
4 12 7 8 15 16 5 13 14 6 11 3 10 9 2
4 12 5 8 15 13 10 14 9 16 7 11 6 3 2
4 13 10 9 15 11 5 14 16 8 2 3 6 7 12
4 14 7 5 16 9 12 15 6 11 10 8 13 3 2
5 6 14 10 16 3 13 7 9 15 11 12 8 2 4
5 6 16 7 13 12 8 11 15 10 4 2 14 3 9
5 7 6 15 12 13 9 8 14 11 16 10 3 4 2
5 7 13 14 16 2 10 9 12 6 15 8 3 11 4
5 7 10 8 13 9 15 12 11 6 16 4 14 2 3
5 7 15 11 16 10 3 14 8 12 6 4 9 13 2
5 8 4 13 14 11 12 10 16 9 15 7 2 3 6
5 8 15 7 14 12 6 11 16 9 3 10 4 13 2
5 8 12 10 15 3 14 11 6 16 9 13 7 2 4
5 8 12 10 16 7 9 13 3 15 11 6 14 4 2
5 8 12 13 16 4 9 14 10 15 3 7 2 6 11
5 9 16 7 14 4 12 11 15 8 3 13 2 10 6
5 9 8 11 15 3 16 13 6 12 14 10 7 2 4
5 9 11 7 15 4 16 14 10 6 12 3 13 2 8
5 10 14 3 12 8 15 11 16 2 9 7 13 4 6
5 10 8 11 15 4 14 12 16 3 13 6 2 7 9
5 12 7 8 15 11 9 13 10 4 16 2 14 6 3
5 12 7 3 16 14 10 15 11 8 9 4 13 6 2
5 13 11 6 16 9 7 14 8 4 12 3 15 10 2
5 13 9 15 16 7 2 14 3 12 11 4 10 8 6
6 7 5 13 12 15 9 8 16 11 14 10 2 4 3
6 7 13 4 16 12 9 10 15 5 11 2 14 8 3
6 7 14 15 13 10 2 12 9 16 3 4 8 5 11
6 7 8 15 16 4 11 13 10 14 9 5 2 3 12
6 7 15 10 16 2 11 14 13 4 8 3 5 12 9
6 8 13 16 10 11 3 9 14 2 15 5 7 12 4
6 8 10 16 11 14 2 9 7 12 15 5 13 4 3
6 8 10 13 14 5 11 9 16 3 15 2 7 4 12
6 8 13 4 15 14 7 11 10 16 3 9 12 5 2
6 8 12 14 16 2 9 11 10 5 15 7 3 13 4
6 8 11 7 16 14 5 12 13 15 2 9 4 10 3
6 9 13 5 12 14 8 10 16 2 11 3 15 7 4
6 9 15 7 14 5 11 10 16 3 8 2 12 4 13
6 9 16 4 14 5 13 11 15 2 8 10 7 12 3
6 9 16 13 14 2 7 11 10 3 12 8 5 15 4
6 9 7 11 16 10 8 12 15 14 4 5 2 3 13
6 10 13 7 12 4 15 11 9 2 16 8 14 5 3
6 10 13 14 15 4 5 11 2 16 9 12 7 8 3
6 10 15 4 14 2 16 12 7 9 8 13 11 5 3
6 10 2 9 15 14 11 12 16 8 13 4 5 3 7
6 11 5 14 13 16 2 12 15 10 8 4 3 7 9
6 11 8 12 16 9 5 13 15 10 4 3 2 7 14
6 13 5 9 16 7 11 15 14 2 12 4 3 10 8
6 13 12 10 16 8 2 15 11 7 3 4 5 14 9
6 14 2 8 16 9 12 15 13 7 10 4 5 3 11
7 8 14 12 16 6 4 9 11 13 5 2 10 3 15
7 8 5 14 16 6 11 10 13 9 15 3 4 2 12
7 8 10 16 14 9 3 11 4 12 15 2 13 5 6
7 8 16 15 14 5 2 13 9 10 4 3 6 11 12
7 8 11 4 16 6 15 14 13 2 12 3 10 9 5
7 9 15 5 12 11 8 10 6 16 4 13 14 3 2
7 9 14 5 16 13 3 10 12 11 4 2 15 6 8
7 9 5 16 12 8 10 11 15 6 14 4 2 3 13
7 9 16 12 13 2 8 11 15 4 5 6 3 10 14
7 9 16 14 15 2 4 11 3 13 8 5 12 6 10
7 9 5 3 14 16 13 12 15 11 8 10 6 4 2
7 10 8 14 12 13 3 11 16 9 6 2 5 4 15
7 10 8 14 13 6 9 11 3 16 12 15 4 5 2
7 10 13 5 16 4 12 11 8 3 15 6 14 9 2
7 10 16 9 14 3 8 12 4 13 5 15 6 11 2
7 10 16 15 14 3 2 12 4 13 5 9 6 11 8
7 10 5 6 15 16 8 12 11 9 13 2 14 3 4
7 10 12 3 14 16 5 13 15 8 2 4 11 9 6
7 10 3 4 16 12 15 14 13 9 11 5 8 2 6
7 11 6 2 16 13 12 14 15 10 4 5 8 3 9
7 12 4 10 16 13 5 14 9 6 15 3 8 11 2
7 12 9 8 16 13 2 15 11 10 3 6 4 14 5
7 13 12 9 15 8 3 14 4 6 11 2 16 10 5
8 9 6 13 12 15 4 10 16 11 7 3 5 2 14
8 9 16 11 14 5 4 12 13 7 2 3 6 10 15
8 9 13 16 15 4 2 12 6 14 5 7 3 10 11
8 9 2 5 16 15 12 14 11 13 10 6 7 3 4
8 10 3 12 15 13 6 11 5 16 14 9 7 4 2
8 10 9 4 16 6 14 11 12 2 15 3 13 5 7
8 10 6 16 15 3 9 12 4 14 13 7 5 2 11
8 10 9 13 16 5 6 12 11 2 15 3 4 14 7
8 10 6 12 15 5 11 14 13 7 9 2 3 4 16
8 11 4 7 16 15 6 13 12 5 14 2 9 10 3
8 11 6 7 16 5 14 15 10 13 4 9 2 3 12
8 12 15 11 16 2 3 14 5 9 4 6 7 13 10
8 12 9 5 16 14 3 15 10 6 7 2 11 13 4
8 13 11 5 16 12 2 15 10 7 3 4 9 14 6
9 10 14 5 15 8 6 13 7 11 3 2 16 4 12
9 11 13 5 15 6 8 12 2 16 4 14 10 7 3
9 11 15 7 16 5 4 14 10 6 2 3 8 13 12
9 12 4 2 16 13 11 15 7 14 6 10 8 5 3
9 12 7 6 16 4 13 15 5 11 8 14 2 10 3
9 13 8 6 16 5 10 15 3 12 7 14 4 11 2
9 13 5 7 16 11 6 15 8 3 14 2 10 12 4
9 14 2 10 16 5 11 15 7 8 12 4 6 3 13
10 11 9 7 16 8 6 14 3 15 5 13 4 12 2
10 12 4 7 15 8 11 13 14 5 9 2 6 3 16
10 12 15 5 16 2 7 13 8 6 3 4 11 9 14
10 12 13 3 16 7 6 14 5 9 4 11 8 15 2
10 13 6 11 16 9 2 14 8 4 12 3 5 15 7
10 13 8 5 16 11 4 15 12 3 6 2 7 14 9
10 14 5 2 16 7 13 15 8 6 9 4 12 3 11
11 12 4 9 15 6 10 14 2 16 8 13 3 5 7
11 13 3 5 16 10 9 14 7 15 4 8 6 2 12
11 13 6 7 16 2 12 15 9 3 10 4 5 8 14
12 13 8 3 16 9 6 14 2 7 11 5 15 10 4
13 14 2 4 16 11 7 15 6 5 12 3 10 9 8
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Posted by Charlie
on 2009-02-03 11:49:52 |
computer solution
