Consider numbers like
343 or
2592.
343=(3+4)^3
2592=2^5*9^2.
We have just shown that some numbers stay unchanged when a number of mathematical operators are added /inserted, without changing the order of the digits.
Within our puzzle lets call those numbers expressionist numbers.
Let us limit the set of acceptable mathematical symbols to the following operators:
+, -, *, /, ^, sqrt, !. and any amount of brackets.
My questions:
In the
period between 10 A.D. & 2016 A.D. what years were labeled by expressionist numbers?
How many such years will there be between 2017 A.D. & 9999 A.D.?
Bonus: How about one, two (or more) 5-digit examples?
Part 1:
Here are RPN versions (translated to algebraic further down below). Square root (postfix r) was limited to twice in a row (so as to allow 4th roots but not 8th roots). For the first batch, depth was limited to 20 characters. Later on, for speed, this was reduced to 16. Other than that, only the 15-decimal-digit accuracy of the programming language limited the possibilities. Only the first found solution is given for each value, even if it wouldn't be the simplest.
All that being the case, it may be that others should be on the list, such as any that require more than two square roots applied successively or have intermediate results that exceed the precision of the programming language. While the final result was allowed to be slightly off to allow for rounding error, such was not applied to intermediate results, so if the 4th power of a fourth root, for example, was slightly off, the result was not allowed to be used to take a factorial, though in fact such rounding would probably be legitimate.
24 24r+!
36 3!6*
71 7!1+r
120 12+!0!-!
144 14r+!4!*
145 14!+5!+
216 21+!r6^
240 24!0!+r!*
324 3!2r/4^
343 34+r3!^
354 35!4r-*
355 35!*5-
360 360!-!*
384 3!rr8r*4^
456 45!6-*
693 6!9rr3!^-
715 71-!5-
720 72r0*!-!
721 72+r!!1+
722 72+r!!2+
723 72+r!!3+
724 72+r!!4+
725 72+r!!5+
726 72+r!!6+
727 72+r!!7+
728 72+r!!8+
729 72+r!!9+
733 73!!+3!+
736 736^+
744 74r+r!!4!+
936 9r!r3!^6!+
1285 128^+5*
1288 12+!r8^8-
1296 12+!r9r!^6*
1298 12*9r!r8^+
1434 14r+!!3-4r*
1435 14r+!!3!!+5-
1439 14r-3!!+9r!!+
1440 14r+!!4r*
1441 14r+!!4r*1+
1442 14r+!!4r*2+
1443 14r+!!4r*3+
1444 14r+!!4r+4r*
1445 14r+!!4r*5+
1446 14r+!!4r*6+
1447 14r+!!4r*7+
1448 14r+!!4r*8+
1449 14r+r4!^r9r!!+
1464 14r+!!6!+4!+
1573 15!+73!+*
1679 167!-9r/-
1680 16+!80!+r/
1704 17!+r0!*4!*
1764 17r*6r*4^
1944 19r!rr*4!^4!/
done
from
DefDbl A-Z
Dim crlf$, stack(20), expr$, done, stackSave(20), yr, stackLevel, fact(15), yrLen, ys$
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
fact(0) = 1
For i = 1 To 15
fact(i) = fact(i - 1) * i
Next
For yr = 10 To 2016
' Text1.Text = Text1.Text & Str(yr)
ys$ = LTrim(Str(yr)): yrLen = Len(ys)
expr$ = Left(ys, 1)
done = 0
stack(1) = Val(expr)
stackLevel = 1
addOn 2
Next
Text1.Text = Text1.Text & crlf & " done"
End Sub
Sub addOn(wh)
If done Then Exit Sub
If Len(expr) > 20 Then Exit Sub
DoEvents
' done = 0
For mono = 1 To 2
expSave$ = expr
stack1save = stack(1)
Select Case mono
Case 1 ' square root
If Right(expr, 2) <> "rr" And stack(1) >= 2 Then
stack(1) = Sqr(stack(1)): expr = expr + "r"
If Abs(stack(1) - yr) < 0.00000001 And stackLevel = 1 And wh >= yrLen Then
Text1.Text = Text1.Text & yr & " " & expr & crlf
done = 1
End If
If done = 0 Then addOn wh
End If
Case 2 ' factorial
If stack(1) = Int(stack(1)) And stack(1) >= 0 And stack(1) < 16 And stack(1) <> 1 And stack(1) <> 2 Then
stack(1) = fact(stack(1)): expr = expr + "!"
If Abs(stack(1) - yr) < 0.00000001 And stackLevel = 1 And wh >= yrLen Then
Text1.Text = Text1.Text & yr & " " & expr & crlf
done = 1
End If
If done = 0 Then addOn wh
End If
End Select
expr = expSave
stack(1) = stack1save
Next
If expr = "34+3!+" Then
x = x
End If
If stackLevel > 1 Then
expSave = expr
For op = 1 To 5
stack1save = stack(1)
stack2save = stack(2)
good = 1
Select Case op
Case 1 ' add
stack(1) = stack(2) + stack(1)
expr = expr + "+"
Case 2 ' subtract
stack(1) = stack(2) - stack(1)
expr = expr + "-"
Case 3 ' multiply
stack(1) = stack(2) * stack(1)
expr = expr + "*"
Case 4 ' divide
If stack(1) <> 0 And Abs(stack(1)) < 1E+100 And Abs(stack(1)) > 1E-200 Then
stack(1) = stack(2) / stack(1)
expr = expr + "/"
Else
good = 0
End If
Case 5 ' exponentiate
If stack(2) = 0 Then
If stack(1) = 0 Then
good = 0
Else
stack(1) = 0
End If
ElseIf stack(1) = 0 Or stack(2) = 1 Then
stack(1) = 1
ElseIf stack(2) < 0 Then
If stack(1) = Int(stack(1)) And Abs(stack(1) * Log(Abs(stack(2)))) < 35 Then
stack(1) = Abs(stack(2)) ^ stack(1)
q = Int(stack1save / 2)
r = stack1save - 2 * q
If r = 1 Then stack(1) = -stack(1)
Else
good = 0
End If
Else
If stack(1) < 1E+100 Then
If stack(1) * Log(Abs(stack(2))) < 35 Then
stack(1) = stack(2) ^ stack(1)
Else
good = 0
End If
Else
good = 0
End If
End If
If good Then expr = expr + "^"
End Select
If good Then
For i = 2 To stackLevel - 1
stack(i) = stack(i + 1)
Next
stackLevel = stackLevel - 1
If expr = "25^92^*" Then
x = x
End If
If Abs(stack(1) - yr) < 0.00000001 And stackLevel = 1 And wh >= yrLen Then
Text1.Text = Text1.Text & yr & " " & expr & crlf
done = 1
Else
addOn wh
End If
liftStack
stackLevel = stackLevel + 1
stack(2) = stack2save
stack(1) = stack1save
expr = expSave
End If
Next op
End If
If wh <= yrLen Then
liftStack
stackLevel = stackLevel + 1
saveExpr$ = expr
stack(1) = Val(Mid(ys, wh, 1))
expr = expr + Mid(ys, wh, 1)
addOn (wh + 1)
expr = saveExpr
For i = 2 To stackLevel - 1
stack(i) = stack(i + 1)
Next
stackLevel = stackLevel - 1
End If
' done = 0
Exit Sub
End Sub
Sub liftStack()
For i = stackLevel + 1 To 2 Step -1
stack(i) = stack(i - 1)
Next
End Sub
Remember that no concatenation is used here. Adjacent digits are separate numbers in the RPN in the second column. Lower case r postfixed represents square root. Two !!'s in a row do not represent double factorial but rather iterated factorial, e.g., 3!! = 6! = 720:
RPN Algebraic
24 24r+! (2+sqrt(4))!
36 3!6* 3! * 6
71 7!1+r sqrt(7! + 1)
120 12+!0!-! ((1+2)! - 0!)!
144 14r+!4!* (1+sqr(4))! * 4!
145 14!+5!+ 1+4!+5!
216 21+!r6^ sqrt((2+1)!)^6
240 24!0!+r!* 2*(sqrt(4!+0!))!
324 3!2r/4^ (3! / sqrt(2))^4
343 34+r3!^ sqrt(3+4)^(3!) (ok, it's a weird way)
354 35!4r-* 3*(5!-sqrt(4))
355 35!*5- 3*5!-5
360 360!-!* 3*(6-0!)!
384 3!rr8r*4^ (sqr(sqr(3!)*sqrt(8))^4
456 45!6-* 4*(5!-6)
693 6!9rr3!^- 6!-sqrt(sqrt(9))^(3!)
715 71-!5- (7-1)!-5
720 72r0*!-! (7-(sqrt(2)*0)!)!
(sqrt above was completely unnecessary but it was the first found soln)
721 72+r!!1+ (sqrt(7+2))!!+1
722 72+r!!2+ (sqrt(7+2))!!+2
723 72+r!!3+ (sqrt(7+2))!!+3
724 72+r!!4+ (sqrt(7+2))!!+4
725 72+r!!5+ (sqrt(7+2))!!+5
726 72+r!!6+ (sqrt(7+2))!!+6
727 72+r!!7+ (sqrt(7+2))!!+7
728 72+r!!8+ (sqrt(7+2))!!+8
729 72+r!!9+ (sqrt(7+2))!!+9
733 73!!+3!+ 7+3!!+3!
736 736^+ 7+3^6
744 74r+r!!4!+ sqrt(7+sqrt(4))!!+4!
936 9r!r3!^6!+ sqrt((sqrt(9))!)^(3!) + 6!
1285 128^+5* (1+2^8)*5
1288 12+!r8^8- sqrt((1+2)!)^8 - 8
The above were converted to algebraic manually but
from here on, program RPN2ALGB converted to algebraic
1296 12+!r9r!^6* sqrt((1+2)!)^sqrt(9)!*6
1298 12*9r!r8^+ 1*2+sqrt(sqrt(9)!)^8
1434 14r+!!3-4r* ((1+sqrt(4))!!-3)*sqrt(4)
1435 14r+!!3!!+5- (1+sqrt(4))!!+3!!-5
1439 14r-3!!+9r!!+ 1-sqrt(4)+3!!+sqrt(9)!!
1440 14r+!!4r* (1+sqrt(4))!!*sqrt(4)
For some reason, here in 1440 and some instances below,
the addition of zero at the end was left out (as in 0+ for RPN or +0 algebraic)
1441 14r+!!4r*1+ (1+sqrt(4))!!*sqrt(4)+1
1442 14r+!!4r*2+ (1+sqrt(4))!!*sqrt(4)+2
1443 14r+!!4r*3+ (1+sqrt(4))!!*sqrt(4)+3
1444 14r+!!4r+4r* ((1+sqrt(4))!!+sqrt(4))*sqrt(4)
1445 14r+!!4r*5+ (1+sqrt(4))!!*sqrt(4)+5
1446 14r+!!4r*6+ (1+sqrt(4))!!*sqrt(4)+6
1447 14r+!!4r*7+ (1+sqrt(4))!!*sqrt(4)+7
1448 14r+!!4r*8+ (1+sqrt(4))!!*sqrt(4)+8
1449 14r+r4!^r9r!!+ sqrt(sqrt((1+sqrt(4)))^4!)+sqrt(9)!!
1464 14r+!!6!+4!+ (1+sqrt(4))!!+6!+4!
1573 15!+73!+* (1+5!)*(7+3!)
1679 167!-9r/- 1-(6-7!)/sqrt(9)
1680 16+!80!+r/ (1+6)!/sqrt((8+0!))
1704 17!+r0!*4!* sqrt((1+7!))*0!*4!
1764 17r*6r*4^ (1*sqrt(7)*sqrt(6))^4
1944 19r!rr*4!^4!/ (1*sqrt(sqrt(sqrt(9)!)))^4!/4!
Remember that all infix operatiors take precedence over diadic operators, such as factorial over exponentiation: x^4! is x^(4!).
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Posted by Charlie
on 2016-05-31 07:27:45 |
Computer "solution" part 1 -- no guarantee of completeness
