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six and thirteen (Posted on 2018-03-15) Difficulty: 3 of 5
Part one:

Compose number 6 with each digit using the digit only three times (Ex: 6=2+2+2) Allowed basic operations (+,-,*,/), sqrt and factorial (no others).

Part two:

Get number 13 using only number 2 three times. Allowed basic and not-basic operations.

No Solution Yet Submitted by armando    
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Solution computer solution | Comment 2 of 8 |
Again repurposing code from three fours:

Translating manually from RPN to algebraic notation, the best for 6 are

(0!+0!+0!)!
(1+1+1)!
2+2+2
2*2+2 to give one not in the example
sqrt(33+3)
4+4-sqrt(4)
5+5/5
6+6-6
7-7/7
8-sqrt(sqrt(8+8)) (this one required a higher length limit in RPN)
(sqrt(8+8/8))!   (another interesting one for 8's)
(9-9/sqrt(9))



111
1,1+,1+!
1,1,1++!
222
2,2+,2+
2,2*,2+
2,2/,2+!
2,2,2++
2,2,2*+
2,2,2/+!
333
33,3+v
3,33+v
3,3+,3-!
3,3-,3+!
3,3-,3!+
3,3*,3-
3,3*,3/!
3,3*v,3+
3,3/,3*!
3,3/,3!*
3,3,3-+!
3,3,3--!
3,3,3*v+
3,3,3/*!
3,3,3//!
3,3,3!--
3,3,3!//
3,3!+,3-
3,3!*,3/
3,3!,3-+
3,3!,3/*
3!,3+,3-
3!,3-,3+
3!,3*,3/
3!,3/,3*
3!,3,3-+
3!,3,3--
3!,3,3/*
3!,3,3//
444
4,4+,4v-
4,4,4/-!
4,4,4v-+
4,4,4v/+
4,4v-,4+
4,4v/,4+
4,4v,4--
555
5,5-!,5+
5,5/,5+
5,5/v,5+
5,5,5-!+
5,5,5/+
5,5,5/v+
666
6,6+,6-
6,6-,6+
6,6-!,6*
6,6-v,6+
6,6*,6/
6,6/,6*
6,6/v,6*
6,6,6-+
6,6,6--
6,6,6-!*
6,6,6-!/
6,6,6-v+
6,6,6-v-
6,6,6/*
6,6,6//
6,6,6/v*
6,6,6/v/
777
7,7,7-!-
7,7,7/-
7,7,7/v-
888
8,8+v!,8/!
8,8-!,8+v!
8,8/,8+v!
8,8/v,8+v!
8,8,8+vv-
8,8,8-!+v!
8,8,8/+v!
8,8,8/v+v!
999
9,9+,9v/
9,9,9v/-

Part 2

A version of the program seeking 13 was unsuccessful, but F(7) = 13, where F(n) is the nth Fibonacci number. We can form 7 via

2,2+s,2-
2,2*s,2-
2,2^s,2-
2,2,9//,2-

The first three involve subfactorials, but the last is 2/.2. - 2, where .2. represents .2 with a bar over the 2. So for 13 we use F(2/.2. - 2).


DefDbl A-Z
Dim crlf$, stack(10), concatable(4), fact(15), dfact(-1 To 25), solCt(1000), shortest(1000) As String
Dim stackPtr, wh, pz As String, expstr As String, lvl, subfact As Variant

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

 
 f = 1: fact(0) = f
 For i = 1 To 15
   f = f * i
   fact(i) = f
   Text1.Text = Text1.Text & f & crlf
 Next
 dfact(-1) = 1: dfact(0) = 1
 
 For i = 1 To 25
   f = 1
   If i Mod 2 = 1 Then
     For j = 1 To i Step 2: f = f * j: Next
   Else
     For j = 2 To i Step 2: f = f * j: Next
   End If
   dfact(i) = f
   Text1.Text = Text1.Text & i & Str(f) & crlf
 Next
 
 subfact = Array(1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496)

 For onlydig = 1 To 9
 
 pz0 = String$(3, LTrim(Str(onlydig)))
 
   pz = pz0
   Text1.Text = Text1.Text & pz & crlf
   h$ = pz
'   Do
     stackPtr = 1
     stack(1) = Val(Left(pz, 1)): concatable(1) = 1
     expstr = LTrim(Str(stack(1)))
     wh = 2
     addOn
    
     
 '    permute pz
 '  Loop Until pz = h
 
' For i = 1 To 1000
'   Text1.Text = Text1.Text & i & "  " & shortest(i) & crlf
'   If shortest(i) = "" Then missct = missct + 1
'   If missct > 2 Then Exit For
' Next
 
 Next onlydig

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

Sub addOn()
 ReDim sstack(4)
 ReDim sconcatable(4)
 
 DoEvents
        For typ = 0 To 8
   sstackPtr = stackPtr
   sexpstr$ = expstr
   For i = 1 To stackPtr
     sstack(i) = stack(i)
     sconcatable(i) = concatable(i)
     swh = wh
   Next
          good = 1
          Select Case typ
            Case 0 ' concatenate
              If concatable(stackPtr) And wh < 4 Then
                stack(stackPtr) = 10 * stack(stackPtr) + Val(Mid(pz, wh, 1))
                expstr = expstr + Mid(pz, wh, 1)
                wh = wh + 1
              Else
                good = 0
              End If
            Case 1  ' add (+)
              If stackPtr > 1 Then
                stackPtr = stackPtr - 1
                stack(stackPtr) = stack(stackPtr) + stack(stackPtr + 1)
                expstr = expstr + "+"
                concatable(stackPtr) = 0
              Else
                good = 0
              End If
             
             Case 2   ' subtract (-)
              If stackPtr > 1 Then
                stackPtr = stackPtr - 1
                stack(stackPtr) = stack(stackPtr) - stack(stackPtr + 1)
                expstr = expstr + "-"
                concatable(stackPtr) = 0
              Else
                good = 0
              End If
            Case 3  ' multiply (*)
              If stackPtr > 1 Then
                stackPtr = stackPtr - 1
                stack(stackPtr) = stack(stackPtr) * stack(stackPtr + 1)
                expstr = expstr + "*"
                concatable(stackPtr) = 0
              Else
                good = 0
              End If
            Case 4  ' divide (/)
              If stackPtr > 1 And stack(stackPtr) <> 0 Then
                stackPtr = stackPtr - 1
                stack(stackPtr) = stack(stackPtr) / stack(stackPtr + 1)
                expstr = expstr + "/"
                concatable(stackPtr) = 0
              Else
                good = 0
              End If
            Case 5   ' comma
              If wh < 4 Then
                stackPtr = stackPtr + 1
                stack(stackPtr) = Val(Mid(pz, wh, 1))
                concatable(stackPtr) = 1
                expstr = expstr + "," + Mid(pz, wh, 1)
                wh = wh + 1
              Else
                good = 0
              End If
            Case 6  ' power (^)
              good = 0
  '           If stackPtr > 1 Then
  '            If stack(stackPtr - 1) <> 0 And stack(stackPtr - 1) < 1000 And stack(stackPtr) >= 0 And stack(stackPtr) < 10 And (stack(stackPtr - 1) > 0 Or stack(stackPtr) = Int(stack(stackPtr))) Then
  '             stackPtr = stackPtr - 1
  '             stack(stackPtr) = stack(stackPtr) ^ stack(stackPtr + 1)
  '             expstr = expstr + "^"
  '             concatable(stackPtr) = 0
  '            Else
  '              good = 0
  '            End If
  '           Else
  '             good = 0
  '           End If
            Case 7  ' factorial
              If stack(stackPtr) = Int(stack(stackPtr)) And stack(stackPtr) <= 15 And stack(stackPtr) >= 0 And stack(stackPtr) <> 1 And stack(stackPtr) <> 2 Then
                stack(stackPtr) = fact(stack(stackPtr))
                expstr = expstr + "!"
                concatable(stackPtr) = 0
               Else
                good = 0
              End If
            Case 8  ' radical
              If stack(stackPtr) >= 0 And Right(expstr, 2) <> "vv" Then
                stack(stackPtr) = Sqr(stack(stackPtr))
                expstr = expstr + "v"
                concatable(stackPtr) = 0
               Else
                good = 0
              End If
             Case 9  ' double factorial
              If stack(stackPtr) = Int(stack(stackPtr)) And stack(stackPtr) <= 25 And stack(stackPtr) >= 4 Then
                stack(stackPtr) = dfact(stack(stackPtr))
                expstr = expstr + "(!!)"
                concatable(stackPtr) = 0
               Else
                good = 0
              End If
            Case 10   ' 4/9
              If wh < 4 Then
                stackPtr = stackPtr + 1
                stack(stackPtr) = 4 / 9
                concatable(stackPtr) = 0
                
                expstr = expstr + ",4,9/"
                wh = wh + 1
              Else
                good = 0
              End If
             Case 11  ' subfactorial
              If stack(stackPtr) = Int(stack(stackPtr)) And stack(stackPtr) <= 9 And stack(stackPtr) >= 4 Then
                stack(stackPtr) = subfact(stack(stackPtr))
                expstr = expstr + "s"
                concatable(stackPtr) = 0
               Else
                good = 0
              End If
             End Select
          If good Then
            If stackPtr = 1 And wh = 4 And stack(1) = 6 Then
            
              If stack(stackPtr) <= 1000 And stack(stackPtr) > 0 And Abs(stack(stackPtr) - Int(stack(stackPtr) + 0.5)) < 0.000000001 Then
            '    solCt(stack(stackPtr)) = solCt(stack(stackPtr)) + 1
            '   If Len(expstr$) < Len(shortest(stack(stackPtr))) Or Len(shortest(stack(stackPtr))) = 0 Then
            '      Text1.Text = Text1.Text & mform(stack(stackPtr), "##0") & "    " & expstr & crlf
                  shortest(stack(stackPtr)) = expstr
                  Text1.Text = Text1.Text & expstr & crlf
            '   End If
                
              End If
            End If
            If Len(expstr) < 10 Then
              addOn
            End If
          End If
   wh = swh
   expstr = sexpstr
   stackPtr = sstackPtr
   For i = 1 To stackPtr
     stack(i) = sstack(i)
     concatable(i) = sconcatable(i)
   Next
   
     
        Next typ

End Sub

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

  Posted by Charlie on 2018-03-15 11:55:36
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