perplexus.info :: Numbers : Parenthetical Integer

Parenthetical Integer (Posted on 2018-04-04)

Parentheses can be inserted into the expression

1/2/3/4/5/6/7/8/9/10/11

to produce a large collection of numbers, some of which are integers. What is the quotient of the largest of these integers divided by the smallest?

No Solution Yet Submitted by Danish Ahmed Khan

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computer solution with explanation that could have been used as an analytic solution (spoiler)

| Comment 1 of 2

The program finds successively lower and higher values by using only division. There's lots of unneeded code as it is based on that for four fours, without removing the dross. A key change, besides looking for largest and smallest, is using only division or placing on the stack (For typ = 4 To 5) for every next number in turn.

The findings in RPN show that the first found integer is in fact the lowest, but the highest was replaced 4 times.

77 1,2/,3/,4/,5/,6,7/,8/,9/,A/,B//

2772 1,2/,3/,4/,5,6/,7/,8/,9/,A/,B//

69300 1,2/,3/,4,5/,6/,7/,8/,9/,A/,B//

1108800 1,2/,3,4/,5/,6/,7/,8/,9/,A/,B//

77 1,2,3//,4,5/,6/,7/,8,9///,A,B//

77 1,2,3//,4,5/,6/,7/,8,9/,A//,B//

9979200 1,2,3/,4/,5/,6/,7/,8/,9/,A/,B//

There are other ways of doing 77 and the only way some are showing up again is due to rounding errors that make it look smaller than before.

Translated from RPN and hexadecimal:

77 = 1/2/3/4/5/(6/7/8/9/10/11)

9979200 = 1/(2/3/4/5/6/7/8/9/10/11)

The largest, 9979200, divided by the smallest, 77, is 129600.

Explanation:

In the expression for 77, ultimately 2, 3, 4, 5 and 6 are in the denominator while 7, 8, 9, 10 and 11 are in the numerator, as the latter set are in the denominator of the denominator. Keeping track of the prime factors, only a single 11 and a single 7 remain after cancellation.

The largest is easier to explain, just half the product of all the numbers other than 2. It's 1/4 of 11! as 2 does not appear in the product that is the numerator, and 2 does appear in the denominator.

DefDbl A-Z

Dim crlf$, stack(15), concatable(13), fact(15), dfact(-1 To 25), solCt(90), shortest(90) As String

Dim stackPtr, wh, pz As String, expstr As String, lvl, subfact As Variant

Dim maxval, minval

Private Sub Form_Load()

Form1.Visible = True

Text1.Text = ""

crlf = Chr$(13) + Chr$(10)

pz0$ = "123456789AB"

minval = 999999

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)

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 90

Text1.Text = Text1.Text & i & " " & shortest(i) & crlf

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

End Sub

Sub addOn()

ReDim sstack(13)

ReDim sconcatable(13)

DoEvents

For typ = 4 To 5

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 < 13 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 < 13 Then

stackPtr = stackPtr + 1

stack(stackPtr) = InStr("123456789AB", (Mid(pz, wh, 1)))

concatable(stackPtr) = 1

expstr = expstr + "," + Mid(pz, wh, 1)

wh = wh + 1

Else

good = 0

End If

Case 6 ' power (^)

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 = 13 Then

If (stack(stackPtr) > maxval Or stack(stackPtr) < minval) 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

If stack(stackPtr) > maxval Then maxval = stack(stackPtr)

If stack(stackPtr) < minval Then minval = stack(stackPtr)

' shortest(stack(stackPtr)) = expstr

' End If

End If

If Len(expstr) < 34 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 2018-04-04 10:25:35

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