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 A modified die (Posted on 2015-02-18)
Take a common die (1 to 6 dots).
Assign to each vertex a number corresponding to the product of numbers denoted by the intersecting faces.
Clearly, the sum of those 8 products is 343.

I have modified the die by changing the number of dots on some (or all) of the faces and the new sum of the products is now 1001.

a. (d2) Find the sum of the new numbers (given they are distinct) - unique answer.
b. (d3) What possible sets of 6 distinct positive integers could enable the new sum?
c. (d3) How should the new numbers be distributed on the modified die?
d. (d3) How many distinct "new dice" exist?

 No Solution Yet Submitted by Ady TZIDON No Rating

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 computer solution | Comment 1 of 3
Here are the 44 modified dice that fit the description. Their totals are each 31.

The first column verifies the sum of products as 1001. The next six columns, in order, show, first the lowest face, then the lowest face that's adjacent to the first, then the other three that are adjacent to the first, counterclockwise from above, and the remaining face, which is opposite the first.

`1001     1 2 3 9 10 6    311001     1 2 3 11 8 6    311001     1 2 4 11 7 6    311001     1 2 5 9 8 6    311001     1 2 7 11 4 6    311001     1 2 8 9 5 6    311001     1 2 8 11 3 6    311001     1 2 10 9 3 6    311001     1 3 4 8 9 6    311001     1 3 4 10 7 6    311001     1 3 7 10 4 6    311001     1 3 9 8 4 6    311001     1 4 5 7 8 6    311001     1 4 8 7 5 6    311001     1 2 3 11 4 10    311001     1 2 4 5 9 10    311001     1 2 4 11 3 10    311001     1 2 6 5 7 10    311001     1 2 7 5 6 10    311001     1 2 9 5 4 10    311001     1 3 5 4 8 10    311001     1 3 6 4 7 10    311001     1 3 7 4 6 10    311001     1 3 8 4 5 10    311001     1 2 3 5 8 12    311001     1 2 3 9 4 12    311001     1 2 4 5 7 12    311001     1 2 4 9 3 12    311001     1 2 7 5 4 12    311001     1 2 8 5 3 12    311001     1 3 5 4 6 12    311001     1 3 6 4 5 12    311001     2 3 4 8 9 5    311001     2 3 4 10 7 5    311001     2 3 6 8 7 5    311001     2 3 7 8 6 5    311001     2 3 7 10 4 5    311001     2 3 9 8 4 5    311001     2 3 5 4 8 9    311001     2 3 6 4 7 9    311001     2 3 7 4 6 9    311001     2 3 8 4 5 9    311001     2 3 5 4 6 11    311001     2 3 6 4 5 11    31`

DefDbl A-Z
Dim crlf\$
Function mform\$(x, t\$)
a\$ = Format\$(x, t\$)
If Len(a\$) < Len(t\$) Then a\$ = Space\$(Len(t\$) - Len(a\$)) & a\$
mform\$ = a\$
End Function

Text1.Text = ""
crlf\$ = Chr(13) + Chr(10)
Form1.Visible = True

For a = 1 To 12
For f = a + 1 To 15
For b = a + 1 To 20
For c = b + 1 To 30
For d = b + 1 To 35
For e = b + 1 To 40
If d <> c And d <> e And c <> e And f <> b And f <> c And f <> d And f <> e Then
tot = a * b * c + a * c * d + a * d * e + a * e * b
tot = tot + f * b * c + f * c * d + f * d * e + f * e * b
If tot = 1001 Then
Text1.Text = Text1.Text & tot & "    " & Str(a) & Str(b) & Str(c) & Str(d) & Str(e) & Str(f) & "   " & Str(a + b + c + d + e + f) & crlf
End If
End If
Next
DoEvents
Next
Next
Next
Next
Next
Text1.Text = Text1.Text & "done"

End Sub

 Posted by Charlie on 2015-02-18 14:07:55

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