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| Inder Taneja's discovery (Posted on 2017-05-11) |
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Natural numbers from 0 to 11111 are written in terms of 1 to 9 in two different ways. The first one in increasing order of 1 to 9, and the second one in decreasing order. This is done by using the operations of addition, multiplication, subtraction, potentiation, and division. In both the situations there are no missing numbers, except one, i.e., 10958 in the increasing case.
Examples: 591=123*4+5*6+78-9 7936=123+4^5+6789 6439 = 9 * (8 + 7 + 6) + 5^4 * (3^2 + 1)
Please try to express the missing number obeying the specified rules.
Summarize your results.
Taneja’s paper is here - https://arxiv.org/abs/1302.1479
some computer results
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10958.3317287401 .331728740100516 ,1,2*,3^,4+,5/,6,7+,8/,9+^
represents
10958.3317287401 as an approximation to 10958 which is less than 1/3 unit away from the goal, and the form of Reverse Polish Notation that I'm using to represent the formula is: ,1,2*,3^,4+,5/,6,7+,8/,9+^, which translates into the algebraic:
(((1*2)^3 + 4) / 5) ^ ((6+7)/8 + 9)
which reduces to
2.4 ^ 10.625
I wrote that while the program was still running. Since then closer values have been found:
10957.7327718651 .267228134922334,1,2^,3-,4,5,6,7-,8/-^,9*+ 10957.8525439016 .147456098377006,1,2^,3,4/,5,6,7*,8*,9/-^+ 10957.8525439016 .147456098377006,1,2^,3,4/,5,6,7,8**,9/-^+ 10957.8525439016 .147456098377006,1,2^,3,4/,5,6,7,8*,9/*-^+
Let's examine
10957.8525439016 .147456098377006,1,2^,3,4/,5,6,7,8**,9/-^+
less than 1/6 unit away from the goal.
The RPN, ,1,2^,3,4/,5,6,7,8**,9/-^+ translates to
1^2 + (3/4) ^ (5 - 6*7*8/9)
= 1 + .75 ^ (-32.333...) ~= 10957.8525439016
Then newer results:
10958.1155551728 .115555172767927,1,2,3/+,4+,5^,6*,7-,8,9/^
or
((1 + 2/3 + 4)^5 * 6 - 7) ^ (8/9) ~= 10958.1155551728
This is the best one found.
The whole output giving successively better approximations follows, with the difference from 10958 shown as the second number on each line, with the first number being the approximation itself.
4480 6478,1,2*,3*,4*,5*,6*,7*,8*,9/ 5023 5935,1,2*,3*,4*,5*,6*,7*,8-,9- 5041 5917,1,2*,3*,4*,5*,6*,7*,8-,9+ 5057 5901,1,2*,3*,4*,5*,6*,7*,8+,9+ 5670 5288,1,2*,3*,4*,5*,6*,7*,8/,9* 5670 5288,1,2*,3*,4*,5*,6*,7*,8,9// 5695 5263,1,2*,3*,4*,5*,6*,7-,8*,9- 5713 5245,1,2*,3*,4*,5*,6*,7-,8*,9+ 6345 4613,1,2*,3*,4*,5*,6*,7-,8-,9* 6489 4469,1,2*,3*,4*,5*,6*,7-,8+,9* 12121 1163,1,2*,3*,4*,5*,6*,7-,8,9+* 10791 167,1,2*,3*,4*,5*,6*,7,8+*,9- 10809 149,1,2*,3*,4*,5*,6*,7,8+*,9+ 10944 14,1,2*,3*,4*,5*,6-,7,89+* 10962 4,1,2*,3*,4*,5*,6+,78,9+* 10959.7327718651 1.73277186507767,1,2*,3*,4-,5,6,7*,8/+^,9* 10959.7327718651 1.73277186507767,1,2*,3*,4-,5,6,7,8/*+^,9* 10958.4 .399999999997817,1,2*,3*,4^,5/,6*,7*,8,9*+ 10958.4 .399999999997817,1,2*,3*,4^,5,6//,7*,8,9*+ 10958.3317287401 .331728740100516,1,2*,3^,4+,5/,6,7+,8/,9+^ 10958.3317287401 .331728740100516,1,2^,3*,4*,5/,6,7+,8/,9+^ 10957.7327718651 .267228134922334,1,2^,3-,4,5,6,7-,8/-^,9*+ 10957.8525439016 .147456098377006,1,2^,3,4/,5,6,7*,8*,9/-^+ 10957.8525439016 .147456098377006,1,2^,3,4/,5,6,7,8**,9/-^+ 10957.8525439016 .147456098377006,1,2^,3,4/,5,6,7,8*,9/*-^+ 10958.1155551728 .115555172767927,1,2,3/+,4+,5^,6*,7-,8,9/^
done DefDbl A-Z Dim crlf$, nlist$, nptr, closest, closelist$, lst$, stack(11), stacklvl
Private Sub Form_Load() Form1.Visible = True Text1.Text = "" crlf = Chr$(13) + Chr$(10) nlist = "123456789" nptr = 1 stacklvl = 0 lst = "" addOn Text1.Text = Text1.Text & crlf & " done" End Sub
Sub addOn() DoEvents lin = Len(lst) If stacklvl = 1 Then If nptr = 10 Then If Abs(10958 - stack(1)) <= Abs(closest - 10958) Then Text1.Text = Text1.Text & stack(1) & Str(Abs(10958 - stack(1))) Text1.Text = Text1.Text & lst & crlf closest = stack(1) closelist = lst End If End If End If If stacklvl >= 2 Then y = stack(2): x = stack(1) stack(1) = x * y For i = 2 To 10 stack(i) = stack(i + 1) Next stacklvl = stacklvl - 1 lst = lst + "*" addOn For i = 11 To 3 Step -1 stack(i) = stack(i - 1) Next stack(2) = y: stack(1) = x stacklvl = stacklvl + 1 lst = Left(lst, Len(lst) - 1) y = stack(2): x = stack(1) stack(1) = y - x For i = 2 To 10 stack(i) = stack(i + 1) Next stacklvl = stacklvl - 1 lst = lst + "-" addOn For i = 11 To 3 Step -1 stack(i) = stack(i - 1) Next stack(2) = y: stack(1) = x stacklvl = stacklvl + 1 lst = Left(lst, Len(lst) - 1) y = stack(2): x = stack(1) stack(1) = x + y For i = 2 To 10 stack(i) = stack(i + 1) Next stacklvl = stacklvl - 1 lst = lst + "+" addOn For i = 11 To 3 Step -1 stack(i) = stack(i - 1) Next stack(2) = y: stack(1) = x stacklvl = stacklvl + 1 lst = Left(lst, Len(lst) - 1) If Abs(stack(1)) > 1E-100 Then y = stack(2): x = stack(1) stack(1) = y / x For i = 2 To 10 stack(i) = stack(i + 1) Next stacklvl = stacklvl - 1 lst = lst + "/" addOn For i = 11 To 3 Step -1 stack(i) = stack(i - 1) Next stack(2) = y: stack(1) = x stacklvl = stacklvl + 1 lst = Left(lst, Len(lst) - 1) End If nflag = 0 If stack(2) = 0 And stack(1) > 0 Then good = 1 Else good = 0 If stack(2) < 0 And stack(1) = Int(stack(1)) Then q = Int(x / 2): r = x - 2 * q If r = 0 Then nflag = 1 good = 1 End If If Abs(stack(2)) > 0 Then If stack(1) * Log(Abs(stack(2))) / Log(10) < 50 Then good = 1 If good Then y = stack(2): x = stack(1) xodd = 0: xeven = 0 If x = Int(x) And x >= 0 Then If stack(2) <> 0 Then good = 0 If good = 0 Then If stack(1) * Log(Abs(stack(2))) / Log(10) < 50 Then good = 1 If good Then q = Int(x / 2): r = x - 2 * q If r = 1 Then xodd = 1 Else xeven = 1 End If End If End If If y >= 0 Or xodd Or xeven Then If y < 0 Then If xodd Then stack(1) = (-Abs(y)) ^ x ElseIf xeven Then stack(1) = (Abs(y)) ^ x End If Else stack(1) = y ^ x End If For i = 2 To 10 stack(i) = stack(i + 1) Next stacklvl = stacklvl - 1 lst = lst + "^" addOn For i = 11 To 3 Step -1 stack(i) = stack(i - 1) Next stack(2) = y: stack(1) = x stacklvl = stacklvl + 1 lst = Left(lst, Len(lst) - 1) End If End If End If ' stacklvl>=2 If Len(lst) <> lin Then xx = xx End If If nptr < 10 Then For i = 11 To 2 Step -1 stack(i) = stack(i - 1) Next stack(1) = Val(Mid(nlist, nptr, 1)) nptr = nptr + 1 stacklvl = stacklvl + 1 lst = lst + "," + Mid(nlist, nptr - 1, 1) addOn For i = 1 To 10 stack(i) = stack(i + 1) Next nptr = nptr - 1 stacklvl = stacklvl - 1 lst = Left(lst, Len(lst) - 2) If Len(lst) <> lin Then xx = xx End If
If Len(lst) > 0 And InStr(nlist, Right(lst, 1)) > 0 Then stack(1) = 10 * stack(1) + Val(Mid(nlist, nptr, 1)) lst = lst + Mid(nlist, nptr, 1) nptr = nptr + 1 addOn nptr = nptr - 1 stack(1) = (stack(1) - Val(Mid(nlist, nptr, 1))) lst = Left(lst, Len(lst) - 1) End If If Len(lst) <> lin Then xx = xx End If End If If Len(lst) <> lin Then xx = xx End If End Sub
The lines that say xx=xx were for placing halts (breakpoints) placed into the program to see where the lst string (the RPN sequence) was losing positions during debugging.
By the way,
for the downward version for the same number, the program finds exact values:
,9,8*,76*,5-,4,3*+,2*,1* ,9,8*,76*,5-,4,3*+,2*,1/ ,9,8*,76*,5-,4,3*+,2*,1^ ,9,8*,76*,5-,4,3*+,2,1** ,9,8*,76*,5-,4,3*+,2,1/* ,9,8*,76*,5-,4,3*+,2,1^* ,9,8*,76*,5,4,3*--,2*,1* ,9,8*,76*,5,4,3*--,2*,1/ ,9,8*,76*,5,4,3*--,2*,1^ ,9,8*,76*,5,4,3*--,2,1** ,9,8*,76*,5,4,3*--,2,1/* ,9,8*,76*,5,4,3*--,2,1^* ,9,8,7*,65*+,4+,3*,2-,1+ ,9,8,7*,65*+,4+,3*,2,1-- ,9,8,7*,65*,4++,3*,2-,1+ ,9,8,7*,65*,4++,3*,2,1-- ,9,8,7,6*-,5,4,3^*,2+*-,1+ ,9,8,7,6*-,5,4,3^*,2+*,1-- ,9,8,7,6,5,4*-,3^+*,2/-,1+ ,9,8,7,6,5,4*-,3^+*,2/,1-- ,9,8,7,6,5,4*-,3^+,2/*-,1+ ,9,8,7,6,5,4*-,3^+,2/*,1-- ,9,8,7,65**+,4+,3*,2-,1+ ,9,8,7,65**+,4+,3*,2,1-- ,9,8,7,65**,4++,3*,2-,1+ ,9,8,7,65**,4++,3*,2,1-- ,9,8,76**,5-,4,3*+,2*,1* ,9,8,76**,5-,4,3*+,2*,1/ ,9,8,76**,5-,4,3*+,2*,1^ ,9,8,76**,5-,4,3*+,2,1** ,9,8,76**,5-,4,3*+,2,1/* ,9,8,76**,5-,4,3*+,2,1^* ,9,8,76**,5,4,3*--,2*,1* ,9,8,76**,5,4,3*--,2*,1/ ,9,8,76**,5,4,3*--,2*,1^ ,9,8,76**,5,4,3*--,2,1** ,9,8,76**,5,4,3*--,2,1/* ,9,8,76**,5,4,3*--,2,1^* ,9,8,76*,5,4+*,3+,2*+,1- ,9,8,76*,5,4+*,3+,2*,1-+ ,9,8,76,5,4+**,3+,2*+,1- ,9,8,76,5,4+**,3+,2*,1-+ ,9,876,5*,4,3,2/-*+,1- ,9,876,5*,4,3,2/-*,1-+ ,9,876,5,4,3,2/-**+,1- ,9,876,5,4,3,2/-**,1-+
Translating some of these into algebraic form:
,9,8*,76*,5-,4,3*+,2,1^* (9*8*76 - 5 + 4*3) * 2^1
,9,8*,76*,5,4,3*--,2*,1* (9*8*76 - (5 - 4*3)) * 2 * 1
,9,8,7*,65*,4++,3*,2-,1+ (9 + 8*7*65 + 4) * 3 - 2 + 1 (Taneja's chosen decreasing formula)
,9,8,7,6*-,5,4,3^*,2+*-,1+ 9 - (8 - 7 * 6) * (5 * 4^3 + 2) + 1
,9,8,7,6,5,4*-,3^+,2/*-,1+ 9 - 8 * (7 + (6 - 5*4)^3)/2 + 1
,9,876,5,4,3,2/-**,1-+ 9 + 876 * 5 * (4 - 3/2) - 1
Edited on May 11, 2017, 2:10 pm
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Posted by Charlie
on 2017-05-11 14:08:31 |
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