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 The five 3īs (Posted on 2008-07-07)
This is a real story. A long long time ago, based on the "Four fours" problem, I wondered if I could do the same using exactly five 3īs, the restrictions a bit tighter: I could use only the 4 basic math operations, exponentiation, factorial, and all parentheses I may need. Besides, I didnīt disallow me to join two "3īs" to make "33".

Using this, and only this, I succeeded in writing expressions for all integers from 0 to 100.

To narrow your work, since a great number of integers can be easily obtained, can you find expressions for 47, 50, 56, 58, 64, 70, 71, 73, 74, 76, 77, 85, 88, 94, and 95?

 See The Solution Submitted by pcbouhid Rating: 4.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 geeky computer solution for some values (spoilers present) | Comment 7 of 35 |

The solution is for those mentioned values that don't require exponentiation.

The list shows the RPN form, the algebraic form and the value:

`3,3!!,3!/+,3/,3!+           (3+3!!/3!)/3+3!              47 33,3/,3!,3!*+               33/3+3!*3!                   47 `
`3,3!*,3*,3!,3/+             3*3!*3+3!/3                  56 3,3!,3**,3!,3/+             3*3!*3+3!/3                  56 3,3!+,3!*,3!,3/+            (3+3!)*3!+3!/3               56 3,3*,3!*,3!,3/+             3*3*3!+3!/3                  56 3,3,3!**,3!,3/+             3*3*3!+3!/3                  56 3,333+,3!/                  (3+333)/3!                   56 333,3+,3!/                  (333+3)/3!                   56 `
`3,3!*,3!!,3!,3*/+           3*3!+3!!/(3!*3)              58 3,3!*,3!!,3!/,3/+           3*3!+(3!!/3!)/3              58 3,3!*,3!!,3,3!*/+           3*3!+3!!/(3*3!)              58 3,3!*,3!!,3/,3!/+           3*3!+(3!!/3)/3!              58 `
`33,3!*,3!-,3/               (33*3!-3!)/3                 64 `
`3,3!,3!*,3!*+,3/            (3+3!*3!*3!)/3               73 3,3!,3!,3!**+,3/            (3+3!*3!*3!)/3               73 3,3/,3!,3!,3!+*+            3/3+3!*(3!+3!)               73 3,3/,3!,3!+,3!*+            3/3+(3!+3!)*3!               73 33,3!!,3!,3*/+              33+3!!/(3!*3)                73 33,3!!,3!/,3/+              33+(3!!/3!)/3                73 33,3!!,3,3!*/+              33+3!!/(3*3!)                73 33,3!!,3/,3!/+              33+(3!!/3)/3!                73 `
`3,3!!,3!,3+/,3!-+           3+3!!/(3!+3)-3!              77 3,3!!,3!,3+/+,3!-           3+3!!/(3!+3)-3!              77 3,3!!,3,3!+/,3!-+           3+3!!/(3+3!)-3!              77 3,3!!,3,3!+/+,3!-           3+3!!/(3+3!)-3!              77 3,3!!,3,3*/,3!-+            3+3!!/(3*3)-3!               77 3,3!!,3,3*/+,3!-            3+3!!/(3*3)-3!               77 3,3!!,3/,3/,3!-+            3+(3!!/3)/3-3!               77 3,3!!,3/,3/+,3!-            3+(3!!/3)/3-3!               77 3,3!,3!!,3!,3+/--           3-(3!-3!!/(3!+3))            77 3,3!-,3!!,3!,3+/+           3-3!+3!!/(3!+3)              77 3,3!,3!!,3,3!+/--           3-(3!-3!!/(3+3!))            77 3,3!-,3!!,3,3!+/+           3-3!+3!!/(3+3!)              77 3,3!,3!!,3,3*/--            3-(3!-3!!/(3*3))             77 3,3!-,3!!,3,3*/+            3-3!+3!!/(3*3)               77 3,3!,3!!,3/,3/--            3-(3!-(3!!/3)/3)             77 3,3!-,3!!,3/,3/+            3-3!+(3!!/3)/3               77 `
`3,3!!,3/,3!+,3/+            3+(3!!/3+3!)/3               85 3,3!,3!!,3/+,3/+            3+(3!+3!!/3)/3               85 `

To find the RPN (vb 5.0):

Private Sub cmdStart_Click()
FontTransparent = False
Open "five3s.txt" For Output As #2
avail = 5
stackMax = -1
cm\$ = ""
place
CurrentX = 1: CurrentY = 1: Print "end"
Close
End Sub

Sub place()
CurrentX = 1: CurrentY = 1: Print cm\$
DoEvents
If Len(cm\$) = 0 Then
cm\$ = "3"
avail = 4
stack(0) = 3
stackMax = 0
place
stackMax = -1
cm\$ = ""
avail = 5
Else
If avail > 0 Then
If Right\$(cm\$, 1) = "3" Then
cm\$ = cm\$ + "3"
stack(0) = 10 * stack(0) + 3
avail = avail - 1
place
cm\$ = Left\$(cm\$, Len(cm\$) - 1)
stack(0) = stack(0) \ 10
avail = avail + 1
End If
cm\$ = cm\$ + ",3"
For i = stackMax + 1 To 1 Step -1
stack(i) = stack(i - 1)
Next i
stack(0) = 3
stackMax = stackMax + 1
avail = avail - 1
place
cm\$ = Left\$(cm\$, Len(cm\$) - 2)
stack(0) = stack(0) \ 10
avail = avail + 1
stackMax = stackMax - 1
For i = 0 To stackMax
stack(i) = stack(i + 1)
Next
End If
If stackMax > 0 Then
cm\$ = cm\$ + "*"
hs0 = stack(0): hs1 = stack(1)
stackMax = stackMax - 1
stack(0) = stack(0) * stack(1)
For i = 1 To stackMax
stack(i) = stack(i + 1)
Next
place
cm\$ = Left\$(cm\$, Len(cm\$) - 1)
stackMax = stackMax + 1
For i = stackMax To 1 Step -1
stack(i) = stack(i - 1)
Next i
stack(0) = hs0: stack(1) = hs1

If stack(0) <> 0 Then
cm\$ = cm\$ + "/"
hs0 = stack(0): hs1 = stack(1)
stackMax = stackMax - 1
stack(0) = stack(1) / stack(0)
For i = 1 To stackMax
stack(i) = stack(i + 1)
Next
place
cm\$ = Left\$(cm\$, Len(cm\$) - 1)
stackMax = stackMax + 1
For i = stackMax To 1 Step -1
stack(i) = stack(i - 1)
Next i
stack(0) = hs0: stack(1) = hs1
End If

cm\$ = cm\$ + "+"
hs0 = stack(0): hs1 = stack(1)
stackMax = stackMax - 1
stack(0) = stack(1) + stack(0)
For i = 1 To stackMax
stack(i) = stack(i + 1)
Next
place
cm\$ = Left\$(cm\$, Len(cm\$) - 1)
stackMax = stackMax + 1
For i = stackMax To 1 Step -1
stack(i) = stack(i - 1)
Next i
stack(0) = hs0: stack(1) = hs1

cm\$ = cm\$ + "-"
hs0 = stack(0): hs1 = stack(1)
stackMax = stackMax - 1
stack(0) = stack(1) - stack(0)
For i = 1 To stackMax
stack(i) = stack(i + 1)
Next
place
cm\$ = Left\$(cm\$, Len(cm\$) - 1)
stackMax = stackMax + 1
For i = stackMax To 1 Step -1
stack(i) = stack(i - 1)
Next i
stack(0) = hs0: stack(1) = hs1

If (stack(0) < 18 And stack(0) > 2 Or stack(0) = 0) And (stack(0) - Int(stack(0)) < 0.0000001 Or stack(0) - Int(stack(0) + 0.5) > 0.9999999) Then
hs0 = stack(0)
cm\$ = cm\$ + "!"
stack(0) = fact(stack(0))
place
stack(0) = hs0
cm\$ = (Left\$(cm\$, Len(cm\$) - 1))
End If
End If
If avail = 0 And stackMax = 0 Then
If stack(0) - Int(stack(0)) < 0.0000001 Or stack(0) - Int(stack(0) + 0.5) > 0.9999999 Then
s0 = Int(stack(0) + 0.5)
If s0 >= 0 And s0 <= 100 Then
printVal = Right("   " + Str(s0), 3)
Print #2, printVal, cm\$
End If
End If
End If
End If
End Sub
Function fact(x)
f = 1
For i = 1 To x
f = f * i
Next
fact = f
End Function

and to convert to algebraid notation (QB):

DECLARE SUB checkParen (s\$, ops\$)
OPEN "five3s.txt" FOR INPUT AS #1
OPEN "five3sal.TXT" FOR OUTPUT AS #2
DO
LINE INPUT #1, l\$
v\$ = LEFT\$(l\$, 14): l\$ = MID\$(l\$, 15)
v = VAL(v\$)
SELECT CASE (v)
CASE 47, 50, 56, 58, 64, 70, 71, 73, 74, 76, 77, 85, 88, 94, 95
stackLevel = 0

bld\$ = ""
FOR i = 1 TO LEN(l\$)
c\$ = MID\$(l\$, i, 1)
ix = INSTR("0123456789", c\$)
IF ix THEN
bld\$ = bld\$ + c\$
stkRaise = 1
ELSE
IF stkRaise THEN
stackLevel = stackLevel + 1
stack\$(stackLevel) = bld\$
stkRaise = 0
bld\$ = ""
END IF
SELECT CASE c\$
CASE "+"
stackLevel = stackLevel - 1
stack\$(stackLevel) = stack\$(stackLevel) + "+" + stack\$(stackLevel + 1)
CASE "-"
h2\$ = stack\$(stackLevel)
checkParen h2\$, "+-"
stackLevel = stackLevel - 1
h1\$ = stack\$(stackLevel)
stack\$(stackLevel) = h1\$ + "-" + h2\$
CASE "*"
h2\$ = stack\$(stackLevel)
checkParen h2\$, "+-"
stackLevel = stackLevel - 1
h1\$ = stack\$(stackLevel)
checkParen h1\$, "+-/"
stack\$(stackLevel) = h1\$ + "*" + h2\$
CASE "/"
h2\$ = stack\$(stackLevel)
checkParen h2\$, "+-/*"
stackLevel = stackLevel - 1
h1\$ = stack\$(stackLevel)
checkParen h1\$, "+-/"
stack\$(stackLevel) = h1\$ + "/" + h2\$
CASE "^"
h2\$ = stack\$(stackLevel)
checkParen h2\$, "+-*/"
stackLevel = stackLevel - 1
h1\$ = stack\$(stackLevel)
checkParen h1\$, "+-*/"
stack\$(stackLevel) = h1\$ + "^" + h2\$
CASE "!"
h2\$ = stack\$(stackLevel)
checkParen h2\$, "+-*/"
stack\$(stackLevel) = h2\$ + "!"
END SELECT
END IF
NEXT
IF stackLevel <> 1 THEN
REM
END IF
PRINT l\$, stack\$(1), v: PRINT #2, l\$, stack\$(1), v
END SELECT
LOOP UNTIL EOF(1)
CLOSE

SUB checkParen (s\$, ops\$)
pLevel = 0
flag = 0
FOR i = 1 TO LEN(s\$)
SELECT CASE MID\$(s\$, i, 1)
CASE "("
pLevel = pLevel + 1
CASE ")"
pLevel = pLevel - 1
CASE ELSE
IF pLevel = 0 AND INSTR(ops\$, MID\$(s\$, i, 1)) THEN flag = 1: EXIT FOR
END SELECT
NEXT i
IF flag THEN s\$ = "(" + s\$ + ")"
END SUB

 Posted by Charlie on 2008-07-07 14:47:24

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