perplexus.info :: Numbers : six and thirteen

six and thirteen (Posted on 2018-03-15)

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

No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)

computer solution

| Comment 2 of 9 |

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

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

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

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

If Len(expstr) < 10 Then

addOn

End If

wh = swh

expstr = sexpstr

stackPtr = sstackPtr

For i = 1 To stackPtr

stack(i) = sstack(i)

concatable(i) = sconcatable(i)

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

Please log in:

Login:
Password:
Remember me:

Sign up! \| Forgot password

Search:

Search body:

Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (7)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info