The following list assumes exactly three 4's are needed.
The subfactorial was used only if no other way resulted in a given number. Other than that, the way of achieving a given number that had the shortest RPN representation is used. In most instances that is the one with the fewest operations, but those that used 4 and 44, rather than three separate 4's, had a slightly better chance, with only one comma instead of two. Those that used double-factorial (!!) were penalized by taking up 4 bytes for this one operation rather than one.
On the RPN side v represents square root and s represents subfactorial; both are postfixed, just like factorial itself, in RPN. As mentioned, double factorial is represented in both RPN and algebraic notation as (!!), while !! would represent the factorial of a factorial. The notation .4. represents .4 with a dot over the 4 to represent repetition; it's listed on the RPN side as ,4,9/
RPN Algebraic
1 4,4/,4^ (4/4)^4
2 4,4+,4/ (4+4)/4
3 4,4,4/- 4-4/4
4 4,4+,4- 4+4-4
5 4,4/,4+ 4/4+4
6 4,4+,4v- 4+4-sqrt(4)
7 4,4!+,4/ (4+(4)!)/4
8 4,4*,4*v sqrt((4*4*4))
9 4,4/,4(!!)+ 4/4+(4)(!!)
10 4,4+,4v+ 4+4+sqrt(4)
11 44,4/ 44/4
12 4,4+,4+ 4+4+4
13 4!,4v+,4v/ ((4)!+sqrt(4))/sqrt(4)
14 4,4*,4v- 4*4-sqrt(4)
15 4,4/,4+(!!) ((4/4+4))(!!)
16 4,4+,4v* (4+4)*sqrt(4)
17 4!,4(!!)(!!)+,4!/ ((4)!+((4)(!!))(!!))/(4)!
18 4,4*,4v+ 4*4+sqrt(4)
19 4(!!),4,9/+,4,9// (4(!!)+.4.)/(.4.)
20 44,4!- 44-(4)!
21 4!,4!,4(!!)/- (4)!-(4)!/(4)(!!)
22 44,4v/ 44/sqrt(4)
23 4!,4,4/- (4)!-4/4
24 4,4+,4-! ((4+4-4))!
25 4,4/,4!+ 4/4+(4)!
26 4,4!+,4v- 4+(4)!-sqrt(4)
27 4!,4!,4(!!)/+ (4)!+(4)!/(4)(!!)
28 4,4*v,4!+ sqrt((4*4))+(4)!
29 4!,4-,4s+ 4!-4+!4
30 4,4!+,4v+ 4+(4)!+sqrt(4)
31 4,4!+,4sv+ 4+4!+sqrt(!4)
32 4,4+,4* (4+4)*4
33 4!,4,4,9//+ (4)!+4/(.4.)
34 4,4(!!)*,4v+ 4*(4)(!!)+sqrt(4)
35 4!,4,9/v-,4,9/v/ ((4)!-sqrt((.4.)))/sqrt((.4.))
36 44,4(!!)- 44-(4)(!!)
37 4,4!+,4s+ 4+4!+!4
38 4!,4,9/v/,4v+ (4)!/sqrt((.4.))+sqrt(4)
39 4!,4v+,4,9/v/ ((4)!+sqrt(4))/sqrt((.4.))
40 44,4- 44-4
41 4!,4(!!)!+,4!/v sqrt((((4)!+((4)(!!))!)/(4)!))
42 44,4v- 44-sqrt(4)
43 4,4s-,4sv!(!!)+ 4-!4+(sqrt(!4))!(!!)
44 44,4^vv sqrt(sqrt(44^4))
45 4(!!)(!!),4!-,4(!!)/ (((4)(!!))(!!)-(4)!)/(4)(!!)
46 44,4v+ 44+sqrt(4)
47 4(!!)(!!),4(!!)-,4(!!)/ (((4)(!!))(!!)-(4)(!!))/(4)(!!)
48 44,4+ 44+4
49 4(!!),4(!!)(!!)+,4(!!)/ ((4)(!!)+((4)(!!))(!!))/(4)(!!)
50 4!,4!+,4v+ (4)!+(4)!+sqrt(4)
51 4!,4(!!)(!!)+,4(!!)/ ((4)!+((4)(!!))(!!))/(4)(!!)
52 44,4(!!)+ 44+(4)(!!)
53 44,4s+ 44+!4
54 4,4*v!,4,9// (sqrt((4*4)))!/(.4.)
55 4!,4,9/+,4,9// ((4)!+.4.)/(.4.)
56 4,4!+,4v* (4+(4)!)*sqrt(4)
57 4!,4!+,4s+ 4!+4!+!4
58 4,4!,4,9//+ 4+(4)!/(.4.)
59 4v,4s+,4sv!(!!)+ sqrt(4)+!4+(sqrt(!4))!(!!)
60 4(!!),4^v,4- sqrt((4)(!!)^4)-4
61 4,4sv^,4sv- 4^(sqrt(!4))-sqrt(!4)
62 4(!!),4^v,4v- sqrt((4)(!!)^4)-sqrt(4)
63 4,4!+,4,9// (4+(4)!)/(.4.)
64 4,4*,4* 4*4*4
65 4(!!)!,4sv!!/,4s+ (4(!!)!)/(sqrt(!4)!)! + !4
66 4v,4(!!),4^v+ sqrt(4)+sqrt((4)(!!)^4)
67 4,4sv^,4sv+ 4^(sqrt(!4))+sqrt(!4)
68 44,4!+ 44+(4)!
69 4!,4sv*,4sv- 4!*sqrt(!4)-sqrt(!4)
70 4(!!)!,4!/,4!/ (((4)(!!))!/(4)!)/(4)!
71 4(!!),4(!!)!+,4(!!)/v sqrt((((4)(!!)+((4)(!!))!)/(4)(!!)))
72 4,4!*,4!- 4*(4)!-(4)!
73 4,4sv^,4s+ 4!^sqrt(!4)+!4
74 4!,4sv*,4v+ 4!*sqrt(!4)+sqrt(4)
75 4!,4sv*,4sv+ 4!*sqrt(!4)+sqrt(!4)
76 4,4!,4sv*+ 4+4!*sqrt(!4)
77 4s,4^v,4- sqrt((!4)^4)-4
78 4!,4!,4,9//+ (4)!+(4)!/(.4.)
79 4s,4^v,4v- sqrt((!4)^4)-sqrt(4)
80 4,4!,4-* 4*((4)!-4)
81 4!,4(!!)/,4^ ((4)!/(4)(!!))^4
82 4v,4sv!!,4s/+ sqrt(4)+(sqrt(!4)!)!/!4
83 4v,4s,4^v+ sqrt(4)+sqrt(!4^4)
84 4,4!+,4sv* (4+4!)*sqrt(!4)
85 4,4s,4^v+ 4+sqrt(!4^4)
86 4sv!,4sv!!,4s/+ (sqrt(!4))!+(sqrt(!4)!)!/!4
87 4,4!*,4s- 4*4!-!4
88 44,4v* 44*sqrt(4)
89 4(!!),4s,4^v+ 4(!!)+(!4)^sqrt(4)
90 4,4v+!,4(!!)/ ((4+sqrt(4)))!/(4)(!!)
91 4(!!),4sv!s+,4sv/ (4(!!)+!(sqrt(!4)!))/sqrt(!4)
92 4,4!*,4- 4*4!-4
93 4,4!*,4sv- 4*4!-sqrt(!4)
94 4,4!*,4v- 4*4!-sqrt(4)
95 4(!!)(!!),4-,4/ (4(!!)(!!)-4)/4
96 4,4*v,4!* sqrt(4*4)*4!
97 4,4(!!)(!!)+,4/ (4+4(!!)(!!))/4
98 4,4!*,4v+ 4*4!+sqrt(4)
99 44,4,9// 44/.4.
100 4,4,4!*+ 4+4*4!
Actually, we can go to 132 before getting to an impossible 133 (but the only multifactorial tried continues to be double factorial, no higher) :
101 4,4sv+(!!),4- (4+sqrt(!4))(!!)-4
102 4,4!*,4sv!+ 4*4!+sqrt(!4)!
103 4,4sv+(!!),4v- (4+sqrt(!4))(!!)-sqrt(4)
104 4,4!,4v+* 4*(4!+sqrt(4))
105 4,4!*,4s+ 4*4!+!4
106 4s,4s(!!)+,4s/ (!4+(!4)(!!))/!4
107 4,4sv+(!!),4v+ (4+sqrt(!4))(!!)+sqrt(4)
108 4,4!,4sv+* 4*(4!+sqrt(!4))
109 4,4,4sv+(!!)+ 4+(4+sqrt(!4))(!!)
110 4v,4s+!,4s!/ (sqrt(4)+!4)!/(!4)!
111 4s,4-!,4s- (!4-4)!-!4
112 4,4,4!+* 4*(4+4!)
113 4,4sv+(!!),4(!!)+ (4+sqrt(!4))(!!)+4(!!)
114 4s,4-!,4sv!- (!4-4)!-sqrt(!4)!
115 4,4sv!s*,4s(!!)- 4*!(sqrt(!4)!)-(!4)(!!)
116 4s,4-!,4- (!4-4)!-4
117 4,4s+,4s* (4+!4)*!4
118 4s,4-!,4v- (!4-4)!-sqrt(4)
119 4,4+(!!),4sv!s- (4+4)(!!)-!(sqrt(!4)!)
120 4,4/,4+! (4/4+4)!
121 4v,4s+,4^v sqrt((sqrt(4)+!4)^4)
122 4v,4s,4-!+ sqrt(4)+(!4-4)!
123 4s,4-!,4sv+ (!4-4)!+sqrt(!4)
124 4,4s,4-!+ 4+(!4-4)!
125 4s,4-,4sv^ (!4-4)^sqrt(!4)
126 4s,4-!,4sv!+ (!4-4)!+(sqrt(!4))!
127 4(!!)s,4sv!,4^+v sqrt(!(4(!!))+(sqrt(!4))!^4)
128 4,4^,4v/ 4^4/sqrt(4)
129 4s,4-!,4s+ (!4-4)!+!4
130 4v,4(!!)(!!),4sv/+ sqrt(4)+(4(!!))(!!)/sqrt(!4)
131 4sv!s,4sv-,4v/ (!(sqrt(!4)!)-sqrt(!4))/sqrt(4)
132 44,4sv* 44*sqrt(!4)
133
134 4sv,4sv!s+,4v/
135 4!,4s-,4s* I quit translating these.
136 4!,4sv!*,4(!!)- (translation program didn't take care
137 4s,4sv!s+,4v/ of subfactorial; I could have added
138 4!,4sv!*,4sv!- that feature, but foolishly did
139 manual conversions)
140 4!,4sv!*,4-
141 4!,4sv!*,4sv-
142 4!,4sv!*,4v-
143 4,4s+(!!),4s(!!)/
144 4,4*,4s*
145 4sv!s,4s,4-!-
146 4!,4sv!*,4v+
147 4!,4sv!*,4sv+
148 4,4!,4sv!*+
149
The program is based on one for a preceding puzzle involving permutations of the digits 3, 1 and 4. The reference to permute, and the enclosing loop are commented out as now all three digits are 4.
DefDbl A-Z
Dim crlf$, stack(10), concatable(4), fact(15), dfact(-1 To 25), solCt(100), shortest(100) As String
Dim stackPtr, wh, pz As String, expstr As String, lvl, subfact As Variant
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
pz0 = "444"
f = 1: fact(0) = f
For i = 1 To 15
f = f * i
fact(i) = f
Text1.Text = Text1.Text & f & crlf
Next
dfact(-1) = 1: dfact(0) = 1
For i = 1 To 25
f = 1
If i Mod 2 = 1 Then
For j = 1 To i Step 2: f = f * j: Next
Else
For j = 2 To i Step 2: f = f * j: Next
End If
dfact(i) = f
Text1.Text = Text1.Text & i & Str(f) & crlf
Next
subfact = Array(1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496)
pz = pz0
Text1.Text = Text1.Text & pz & crlf
h$ = 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 100
Text1.Text = Text1.Text & i & " " & shortest(i) & crlf
Next
Text1.Text = Text1.Text & crlf & " done"
End Sub
Sub addOn()
ReDim sstack(4)
ReDim sconcatable(4)
DoEvents
For typ = 0 To 11
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 (^)
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 Then
If stack(stackPtr) <= 100 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
End If
End If
End If
If Len(expstr) < 34 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
Edited on December 6, 2017, 7:37 pm
Edited on December 7, 2017, 9:01 am
|
|
Posted by Charlie
on 2017-12-06 19:34:47 |
computer solutions (spoilers)
