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Zero Sum Expression (Posted on 2015-03-25) Difficulty: 3 of 5
Define p ⊗ q = p+q-p*q, and find all possible triplets (a, b, c) of integers such that:
a ⊗ ( b ⊗ c) + b ⊗ (c ⊗ a) + c ⊗ (a ⊗ b) = 0

No Solution Yet Submitted by K Sengupta    
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Some Thoughts computer exploration | Comment 1 of 2
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

Private Sub Form_Load()
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
 
 For ptot = 0 To 10000
   For a0 = 0 To ptot / 3
   For b0 = a0 To (ptot - a0) / 2
     For a = -a0 To a0 Step 2 * a0
     For b = -b0 To b0 Step 2 * b0
       cd = ptot - a - b
       For c = -cd To cd Step 2 * cd
       If f(a, f(b, c)) + f(b, f(c, a)) + f(c, f(a, b)) = 0 Then
           Text1.Text = Text1.Text & a & Str(b) & Str(c) & crlf
           DoEvents
       End If: If c = 0 Then Exit For
       Next c: If b = 0 Then Exit For
     Next: If a = 0 Then Exit For
     Next
   Next
   Next
 Next ptot
 
 Text1.Text = Text1.Text & crlf & crlf
 


End Sub

Function f(x, y)
  f = x + y - x * y
End Function

finds only (0,0,0) and (0,2,2), as it shows only ascending absolute values, so permutations of the latter should also work.

  Posted by Charlie on 2015-03-25 15:44:18
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