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 Minimizing n (Posted on 2016-03-23)
Given:
i. a, b, c, d, e, f, g, h are distinct positive integers.
ii. ab+cd=ef+gh=n

Find the minimum value of n.

 See The Solution Submitted by Ady TZIDON No Rating

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 computer confirmation (solution) Comment 2 of 2 |
DefDbl A-Z
Dim crlf\$, used(30)

Form1.Visible = True

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

minimum = 100

For a = 1 To 23
used(a) = 1
For b = a + 1 To 30
used(b) = 1
For c = a + 1 To 29
If used(c) = 0 Then
used(c) = 1
For d = c + 1 To 30
If used(d) = 0 Then
used(d) = 1
For e = a + 1 To 29
If used(e) = 0 Then
used(e) = 1
DoEvents
For f = e + 1 To 30
If used(f) = 0 Then
used(f) = 1
For g = e + 1 To 29
If used(g) = 0 Then
used(g) = 1
For h = g + 1 To 30
If used(h) = 0 Then
used(h) = 1
If a * b + c * d = e * f + g * h Then
If a * b + c * d <= minimum Then
Text1.Text = Text1.Text & Str(a) & Str(b) & Str(c) & Str(d) & Str(e) & Str(f) & Str(g) & Str(h) & "  " & a * b + c * d & crlf
minimum = a * b + c * d
End If
End If
used(h) = 0
End If
Next
used(g) = 0
End If
Next
used(f) = 0
End If
Next
used(e) = 0
End If
Next
used(d) = 0
End If
Next
used(c) = 0
End If
Next
used(b) = 0
Next
used(a) = 0
Next

Text1.Text = Text1.Text & minimum & crlf & " done"

End Sub

finds the minimum is 31, per:

1 2 3 19 4 6 5 7  59
1 2 4 13 3 8 5 6  54
1 2 5 10 3 8 4 7  52
1 2 6 7 3 8 4 5  44
1 4 5 7 2 6 3 9  39
1 5 3 11 2 7 4 6  38
1 6 3 10 2 8 4 5  36
1 6 4 7 2 5 3 8  34
1 7 3 9 2 5 4 6  34
1 7 4 6 2 8 3 5  31

where the last, best, line represents

1*7 + 4*6 = 2*8 + 3*5 = 31

where a had to be the smallest number of all 8
b had to be larger than a (of course)
same with c larger than a
d larger than c
e larger than a
f larger than e
g larger than e
h larger than g

wlog. The rules merely eliminate trivial variations.

 Posted by Charlie on 2016-03-23 13:34:40

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