perplexus.info :: Numbers : Four fours

Four fours (Posted on 2018-02-14)

Arrange 4 fours into various expressions to generate all integer numbers, from 1 to 112 inclusive like:

1 = 44 ÷ 44
2=sqrt 4 - (4 - 4) x 4
3 = (4 + 4 + 4) ÷ 4
4 = 4 + 4 × (4 - 4)
... ...
7= 44/4-4
... ...
19=4!-4-4/4
... ...
100=4*4!+sqrt(4*4)
etc

Allow use of concatenation, +, -, x, /, sqrt, exponentation, factorial,decimal fractions, overline & brackets.

Source: Inspired by Martin Gardner statement.

No Solution Yet Submitted by Ady TZIDON

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minor tweak to three fours program; 7 goal numbers remain unsolved

| Comment 1 of 7

The program at bottom is based on the one for the "three fours" puzzle, except of course it uses four 4's. Also it does not use double factorials or subfactorials, though the setup of those is performed at the beginning of the program--that doesn't do any harm.

As in "three fours", .4. in the algebraic expression column represents .4 with a bar over the 4, or 4/9.

The left column has, separated by spaces, the number represented, how many representations of that number were found using four 4's and the RPN representation of one shortest such formula. The last column shows the algebraic notation representation of the four-4's formula. The formula that is shortest when represented in RPN (or one of those tied for shortest). Seven numbers had zero ways of being produced.

1 1481358 44,44/            44/44
2 152205 44,4+,4!/          (44+4)/(4)!
3 39674 44,4/,4v-v          sqrt((44/4-sqrt(4)))
4 81584 44,4-,4!-v          sqrt((44-4-(4)!))
5 43137 44,4!-,4/           (44-(4)!)/4
6 59911 44,4-,4-v           sqrt((44-4-4))
7 3665 44,4/,4-             44/4-4
8 18470 44,4-,4!+v          sqrt((44-4+(4)!))
9 4807 44,4/,4v-            44/4-sqrt(4)
10 3684 44,4-,4/            (44-4)/4
11 324 44,4,4*v/            44/sqrt((4*4))
12 14979 44,4+,4/           (44+4)/4
13 411 44,4/,4v+            44/4+sqrt(4)
14 1017 4,4+,4+,4v+         4+4+4+sqrt(4)
15 480 44,4/,4+             44/4+4
16 16673 44,4-,4!-          44-4-(4)!
17 762 44,4!+,4/            (44+(4)!)/4
18 2624 44,4v/,4-           44/sqrt(4)-4
19 591 4!,4-,4,4/-          (4)!-4-4/4
20 5149 44,4-,4v/           (44-4)/sqrt(4)
21 1092 44,4v-,4v/          (44-sqrt(4))/sqrt(4)
22 4698 44,4/,4v*           (44/4)*sqrt(4)
23 11072 44,4v+,4v/         (44+sqrt(4))/sqrt(4)
24 151711 44,4+,4!-         44+4-(4)!
25 21694 4,4/,4+,4^v        sqrt((4/4+4)^4)
26 8262 44,4v/,4+           44/sqrt(4)+4
27 2368 4,4,4/-,4!+         4-4/4+(4)!
28 7044 44,4,4*-            44-4*4
29 1413 4,4/,4+,4!+         4/4+4+(4)!
30 3051 4,4+,4*,4v-         (4+4)*4-sqrt(4)
31 52 4,4!+,4/,4!+          (4+(4)!)/4+(4)!
32 4079 44,4!,4v/-          44-(4)!/sqrt(4)
33 124 44,4,9/,4*v/         44/sqrt(((.4.)*4))
34 438 44,4!+,4v/           (44+(4)!)/sqrt(4)
35 102 44,4/,4!+            44/4+(4)!
36 5711 44,4-,4-            44-4-4
37 101 4,4,4,9//+,4!+       4+4/(.4.)+(4)!
38 291 44,4-,4v-            44-4-sqrt(4)
39 35 4,4!+,4,9//,4!-       (4+(4)!)/(.4.)-(4)!
40 1225 44,4,4*v-           44-sqrt((4*4))
41 16 4,4+!,4!+,4!/v        sqrt(((((4+4))!+(4)!)/(4)!))
42 233 44,4-,4v+            44-4+sqrt(4)
43 75 44,4,4/-              44-4/4
44 1120 44,44*v             sqrt((44*44))
45 244 44,4,4/+             44+4/4
46 677 44,4+,4v-            44+4-sqrt(4)
47 511 4!,4,4/-,4!+         (4)!-4/4+(4)!
48 15553 44,4,4*v+          44+sqrt((4*4))
49 834 4,4/,4!+,4!+         4/4+(4)!+(4)!
50 1130 44,4+,4v+           44+4+sqrt(4)
51 23 4!,4!,4,9//,4v/+      (4)!+((4)!/(.4.))/sqrt(4)
52 1001 44,4+,4+            44+4+4
53 88 4,4,9//,44+           4/(.4.)+44
54 2162 4,4v+,4^,4!/        (4+sqrt(4))^4/(4)!
55 87 4!,4,9//,4,4/+        (4)!/(.4.)+4/4
56 552 44,4!,4v/+           44+(4)!/sqrt(4)
57 20 4!,4,4,9//+,4!+       (4)!+4/(.4.)+(4)!
58 90 4,4^,4!-,4/           (4^4-(4)!)/4
59 4 4,4!+,4,9//,4-         (4+(4)!)/(.4.)-4
60 722 44,4,4*+             44+4*4
61 3 4,4!+,4,9//,4v-        (4+(4)!)/(.4.)-sqrt(4)
62 73 4,4*,4*,4v-           4*4*4-sqrt(4)
63 35 4,4^,4-,4/            (4^4-4)/4
64 4158 44,4-,4!+           44-4+(4)!
65 4 4,4,4^+,4/             (4+4^4)/4
66 137 44,4!+,4v-           44+(4)!-sqrt(4)
67 2 4,4!+,4,9//,4+         (4+(4)!)/(.4.)+4
68 214 44,4,4*v!+           44+(sqrt((4*4)))!
69 1 4!,4-,4,9//,4!+        ((4)!-4)/(.4.)+(4)!
70 101 44,4!+,4v+           44+(4)!+sqrt(4)
71 0           
72 2398 44,4+,4!+           44+4+(4)!
73 0           
74 90 4,4!*,4!-,4v+         4*(4)!-(4)!+sqrt(4)
75 1 44,4,9//,4!-           44/(.4.)-(4)!
76 99 4,4!,4-*,4-           4*((4)!-4)-4
77 9 4,4,9//,4^v,4-         sqrt((4/(.4.))^4)-4
78 96 4,4!,4-*,4v-          4*((4)!-4)-sqrt(4)
79 9 4,4,9//,4^v,4v-        sqrt((4/(.4.))^4)-sqrt(4)
80 322 44,4-,4v*            (44-4)*sqrt(4)
81 818 4,4/,4-,4^           (4/4-4)^4
82 19 4,4!,4-*,4v+          4*((4)!-4)+sqrt(4)
83 12 4v,4,4,9//,4^v+       sqrt(4)+sqrt((4/(.4.))^4)
84 55 44,4v*,4-             44*sqrt(4)-4
85 12 4,4,4,9//,4^v+        4+sqrt((4/(.4.))^4)
86 4 44,4v*,4v-             44*sqrt(4)-sqrt(4)
87 2 4,4!+,4,9//,4!+        (4+(4)!)/(.4.)+(4)!
88 345 44,44+               44+44
89 0           
90 60 44,4v*,4v+            44*sqrt(4)+sqrt(4)
91 0          
92 357 44,4v*,4+            44*sqrt(4)+4
93 0           
94 147 4,4!*,4-,4v+         4*(4)!-4+sqrt(4)
95 145 44,4,9//,4-          44/(.4.)-4
96 9398 44,4+,4v*           (44+4)*sqrt(4)
97 289 4,4/,4,4!*+          4/4+4*(4)!
98 271 4,4,4!*+,4v-         4+4*(4)!-sqrt(4)
99 22 44,4,9//,4^vv         sqrt(sqrt((44/(.4.))^4))
100 740 4,4/,4!+,4*         (4/4+(4)!)*4
101 1 44,4,9//,4v+          44/(.4.)+sqrt(4)
102 50 4,4,4!*+,4v+         4+4*(4)!+sqrt(4)
103 1 44,4,9//,4+           44/(.4.)+4
104 308 4,4+,4,4!*+         4+4+4*(4)!
105 16 4,4,9//,4,4!*+       4/(.4.)+4*(4)!
106 20 4,4!,4v+*,4v+        4*((4)!+sqrt(4))+sqrt(4)
107 0          
108 518 4,4,4!+*,4-         4*(4+(4)!)-4
109 0          
110 25 4,4,4!+*,4v-         4*(4+(4)!)-sqrt(4)
111 5 444,4/                444/4
112 349 44,4v*,4!+          44*sqrt(4)+(4)!

DefDbl A-Z

Dim crlf$, stack(10), concatable(5), fact(15), dfact(-1 To 25), solCt(112), shortest(112) 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 = "4444"

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 112

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

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

End Sub

Sub addOn()

ReDim sstack(5)

ReDim sconcatable(5)

DoEvents

For typ = 0 To 10

If typ <> 9 Then

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 < 5 Then

stack(stackPtr) = 10 * stack(stackPtr) + Val(Mid(pz, wh, 1))

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

If expstr = "44,44" Then

xx = xx

End If

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 < 5 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 = 5 Then

If stack(stackPtr) <= 112 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

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)

End If ' not typ=9

Next typ

End Sub

Posted by Charlie on 2018-02-14 10:35:12

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