The shopping trips can be represented by the following seven sets of digits

114 123 213 222 312 321 411 

Each set of three digits represents the number of boxes bought at B, at S and at P respectively. We of course don't know which of the 7! = 5040 possible orders these trips were made in, but we see that Mother bought 4+3+3+2+2+1+1 = 16 boxes at P's Stores to get those 80 cakes.

She also bought 1+2+1+2+1+2+1 = 10 boxes at S's Bakery, giving the answer of 10*4 = 40 cakes.

Not that it was asked, but she bought 1+1+2+2+3+3+4 = 16 boxes at B's Tea Room, for 48 cakes.  That's 168 cakes in all for the month.

We can verify that indeed each day's number of cakes was different:

27, 26, 25, 24, 23, 22 and 21

The answer assumes that Mother bought at least one box from each of the three stores. Allowing her to skip either B or S on a given day would allow many more solutions. The program itself did not guard against skipping P on a given day, but it found no such solutions.

DefDbl A-Z

Dim crlf$, numBoxes(7, 3), totCakes(7), overtotP(7), overtotS(7)

Private Sub Form_Load()

 Form1.Visible = True

 

 

 Text1.Text = ""

 crlf = Chr$(13) + Chr$(10)

 

 setup 1

 

 

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

  

End Sub

Sub setup(trip)

  For b = 1 To 6

   For s = 1 To 6 - b

     p = 6 - b - s

     numBoxes(trip, 1) = b

     numBoxes(trip, 2) = s

     numBoxes(trip, 3) = p

     totCakes(trip) = 3 * b + 4 * s + 5 * p

     overtotP(trip) = overtotP(trip - 1) + 5 * p

     overtotS(trip) = overtotS(trip - 1) + 4 * s

     DoEvents

     If overtotP(trip) <= 80 Then

       good = 1

       For i = 1 To trip - 1

         If totCakes(i) = totCakes(trip) Then good = 0: Exit For

       Next i

       If good Then

         If trip = 7 Then

           If overtotP(trip) = 80 Then

             ReDim nm(7): good = 1

             For tr = 1 To 7

              For i = 1 To 3

               nm(tr) = 10 * nm(tr) + numBoxes(tr, i)

              Next

              If nm(tr) < nm(tr - 1) Then good = 0: Exit For

             Next tr

             If good Then

               For i = 1 To 7

                 Text1.Text = Text1.Text & nm(i) & " "

               Next i

               Text1.Text = Text1.Text & overtotS(trip) & "  " & crlf

             End If

           End If

         Else

           setup trip + 1

         End If

       End If

     End If

     ct = ct + 1

   Next s

  Next b

End Sub

The program could have been more efficient, as the requirement that the representations be in order when treated as 3-digit numbers was added as an afterthought. Ideally it would have been checked as it went along, to prune the recursive tree and saving much time.