**“It’s not that I’m so smart, it’s just that I stay with problems longer.” [Albert Einstein]**

## Abstract

Mr Excel’s challenge of the month of August 2002 stated:

“An accounts receivable department receives a check from a customer for $4,556.92. Upon looking in the accounting system, there are 54 unpaid invoices, ranging from $77.74 to $5,465.45. The payment must be for some exact combination of entire invoices, but we don’t know which invoices are being paid.”

The winning solution was published (external link!) here. Michael Schwimmer presented a nice and elegant solution on his (now retired) website - in German. I translated his version into English because I found it really beautiful. Any errors are mine, I am sure.

## Appendix – cmbCalculate_Click Code

Please read my Disclaimer.

```
Private Sub cmbCalculate_Click()
Dim dGoal As Double
Dim dTolerance As Double
Dim dAmounts() As Double
Dim vResult As Variant
Dim m As Long
Dim n As Long
With Me
dGoal = .Range("B2")
dTolerance = .Range("C2")
ReDim dAmounts(1 To 100)
For m = 2 To 101
If (.Cells(m, 1) <> "") And (IsNumeric(.Cells(m, 1))) Then
dAmounts(m - 1) = .Cells(m, 1)
Else
ReDim Preserve dAmounts(1 To m - 1)
Exit For
End If
Next
ReDim Preserve dAmounts(1 To UBound(dAmounts) - 1)
vResult = Combinations(dAmounts, dGoal, dTolerance)
Application.ScreenUpdating = False
.Range("D3:D65536").ClearContents
.Range(.Cells(3, 4), .Cells(UBound(vResult) + 3, 4)) = vResult
Application.ScreenUpdating = True
End With
End Sub
Function Combinations( _
Elements As Variant, _
Goal As Double, _
Optional Tolerance As Double, _
Optional SoFar As Variant, _
Optional Position As Long) As Variant
Dim i As Long
Dim k As Long
Dim dCompare As Double
Dim dDummy As Double
Dim vDummy As Variant
Dim vResult As Variant
If Not IsMissing(SoFar) Then
'Sum of elements so far
For Each vDummy In SoFar
dCompare = dCompare + vDummy
Next
Else
'Start elements sorted by amount
For i = 1 To UBound(Elements)
For k = i + 1 To UBound(Elements)
If Elements(k) < Elements(i) Then
dDummy = Elements(i)
Elements(i) = Elements(k)
Elements(k) = dDummy
End If
Next
Next
Set SoFar = New Collection
End If
If Position = 0 Then Position = LBound(Elements)
For i = Position To UBound(Elements)
'Add current element
SoFar.Add Elements(i)
dCompare = dCompare + Elements(i)
If Abs(Goal - dCompare) < (0.001 + Tolerance) Then
'Goal achieved
k = 0
ReDim vResult(0 To SoFar.Count - 1, 0)
For Each vDummy In SoFar
vResult(k, 0) = vDummy
k = k + 1
Next
Combinations = vResult
Exit For
ElseIf dCompare < (Goal + 0.001 + Tolerance) Then
'Enough room for another element
'Call recursively starting with next higher amount
vResult = Combinations(Elements, Goal, Tolerance, SoFar, i + 1)
If IsArray(vResult) Then
Combinations = vResult
Exit For
Else
SoFar.Remove SoFar.Count
dCompare = dCompare - Elements(i)
End If
Else
'Amount too high
SoFar.Remove SoFar.Count
Exit For
End If
Next 'Try next higher amount
End Function
```

Please read my Disclaimer.

Accounts_Receivable_Problem.xlsm [27 KB Excel file, open and use at your own risk]