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Inder Taneja's discovery (Posted on 2017-05-11) Difficulty: 3 of 5
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

No Solution Yet Submitted by Ady TZIDON    
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Comments: ( Back to comment list | You must be logged in to post comments.)
A step beyond Taneja's list | Comment 2 of 4 |
(In reply to some computer results by Charlie)

Taneja stopped at 11111. I don't know how far it can go, but 11112 is certainly accessible:

Forward:

11112 0,1,2*,3*,4+,5^,6,7-,8*-,9/
11112 0,1,2*,3,4*-,5^,6,7-*,8+,9/
11112 0,1,2*,3,4*-,5^,6,7-/,8+,9/
11112 0,1,2*,3,4*,5,6+,7*,8*,9*+*
11112 0,1,2*,3,4*,5,6+,7*,8,9**+*
11112 0,1,2*,3,4*,5,6+,7,8**,9*+*
11112 0,1,2*,3,4*,5,6+,7,8*,9**+*
11112 0,1,2*,3,4*,5,6+,7,8,9***+*
11112 0,1,2*,3,4,5+,6,7,89*-*-*
11112 0,1,2-,3-,4*,5,6/,78-*,9*
11112 0,1,2-,3-,4*,5,6/,78-,9**
11112 0,1,2-,3-,4,5,6/,78-**,9*
11112 0,1,2-,3-,4,5,6/,78-*,9**
11112 0,1,2-,3-,4,5,6/,78-,9***
11112 0,1,2+,3+,4+,5^,6,7-,8*-,9/
11112 0,1,2+,3^,456,7+*,8*,9/
11112 0,1,2+,3^,456,7+*,8,9/*
11112 0,1,2+,3^,456,7+,8**,9/
11112 0,1,2+,3^,456,7+,8*,9/*
11112 0,1,2+,3^,456,7+,8,9/**
11112 0,1,2+,3,4++,5^,6,7-,8*-,9/
11112 0,1,2+,3,456,7+,8*+*,9-
11112 0,1,2+,34,5,67*,8-*+,9-
11112 0,1,2+,34,5,67*,8-*,9-+
11112 0,1,2/,3-,4*,5^,6,7-*,8+,9/
11112 0,1,2/,3-,4*,5^,6,7-/,8+,9/
11112 0,1,2^,3/,456,7+,8*,9**
11112 0,1,2^,3/,456,7+,8,9***
11112 0,1,2^,3,456,7+//,8*,9*
11112 0,1,2^,3,456,7+//,8,9**
11112 0,1,2^,3,456,7+/,8//,9*
11112 0,1,2^,3,456,7+/,8/,9//
11112 0,1,2^,3,456,7+/,8,9*//
11112 0,1,2^,3,456,7+,8*//,9*
11112 0,1,2^,3,456,7+,8*/,9//
11112 0,1,2^,3,456,7+,8*,9*//
11112 0,1,2^,3,456,7+,8,9**//
11112 0,1,2,3**,4+,5^,6,7-,8*-,9/
11112 0,1,2,3*+,456+,7,8+,9+*
11112 0,1,2,3*+,456+,7,8,9++*
11112 0,1,2,3*,4+*,5^,6,7-,8*-,9/
11112 0,1,2,3*,4+,5^*,6,7-,8*-,9/
11112 0,1,2,3*,4+,5^+,6+,7-,8+,9/
11112 0,1,2,3*,4+,5^+,6+,7,8--,9/
11112 0,1,2,3*,4+,5^+,6,7-+,8+,9/
11112 0,1,2,3*,4+,5^+,6,7-,8++,9/
11112 0,1,2,3*,4+,5^+,6,7,8--+,9/
11112 0,1,2,3*,4+,5^,6++,7-,8+,9/
11112 0,1,2,3*,4+,5^,6++,7,8--,9/
11112 0,1,2,3*,4+,5^,6+,7-+,8+,9/
11112 0,1,2,3*,4+,5^,6+,7-,8++,9/
11112 0,1,2,3*,4+,5^,6+,7,8--+,9/
11112 0,1,2,3*,4+,5^,6,7-++,8+,9/
11112 0,1,2,3*,4+,5^,6,7-+,8++,9/
11112 0,1,2,3*,4+,5^,6,7-,8*-*,9/
11112 0,1,2,3*,4+,5^,6,7-,8*-,9/*
11112 0,1,2,3*,4+,5^,6,7-,8+++,9/
11112 0,1,2,3*,4+,5^,6,7-,8^-,9/+
11112 0,1,2,3*,4+,5^,6,7,8--++,9/
11112 0,1,2,3*,456++,7,8+,9+*
11112 0,1,2,3*,456++,7,8,9++*
11112 0,1,2,3+-,4*,5,6/,78-*,9*
11112 0,1,2,3+-,4*,5,6/,78-,9**
11112 0,1,2,3+-,4,5,6/,78-**,9*
11112 0,1,2,3+-,4,5,6/,78-*,9**
11112 0,1,2,3+-,4,5,6/,78-,9***
11112 0,1,2,3++,4+,5^,6,7-,8*-,9/
11112 0,1,2,3+,4++,5^,6,7-,8*-,9/
11112 0,1,2,3/-,456,7+*,8*,9*
11112 0,1,2,3/-,456,7+*,8,9**
11112 0,1,2,3/-,456,7+,8**,9*
11112 0,1,2,3,4*-*,5^,6,7-*,8+,9/
11112 0,1,2,3,4*-*,5^,6,7-/,8+,9/
11112 0,1,2,3,4*-,5^*,6,7-*,8+,9/
11112 0,1,2,3,4*-,5^*,6,7-/,8+,9/
11112 0,1,2,3,4*-,5^-,6+,7-,8+,9/
11112 0,1,2,3,4*-,5^-,6+,7,8--,9/
11112 0,1,2,3,4*-,5^-,6,7-+,8+,9/
11112 0,1,2,3,4*-,5^-,6,7-,8++,9/
11112 0,1,2,3,4*-,5^-,6,7,8--+,9/
11112 0,1,2,3,4*-,5^,6--,7-,8+,9/
11112 0,1,2,3,4*-,5^,6--,7,8--,9/
11112 0,1,2,3,4*-,5^,6-,7+-,8+,9/
11112 0,1,2,3,4*-,5^,6-,7+,8--,9/
11112 0,1,2,3,4*-,5^,6-,7,8-+-,9/
11112 0,1,2,3,4*-,5^,6,7-**,8+,9/
11112 0,1,2,3,4*-,5^,6,7-*,8+*,9/
11112 0,1,2,3,4*-,5^,6,7-*,8+,9/*
11112 0,1,2,3,4*-,5^,6,7---,8+,9/
11112 0,1,2,3,4*-,5^,6,7--,8--,9/
11112 0,1,2,3,4*-,5^,6,7-/*,8+,9/
11112 0,1,2,3,4*-,5^,6,7-/,8+*,9/
11112 0,1,2,3,4*-,5^,6,7-/,8+,9/*
11112 0,1,2,3,4*-,5^,6,7-,8+--,9/
11112 0,1,2,3,4*-,5^,6,7-,8^+,9/-
11112 0,1,2,3,4*-,5^,6,7,8----,9/
11112 0,1,2,3,4*,5,6+,7*,8*,9*+**
11112 0,1,2,3,4*,5,6+,7*,8,9**+**
11112 0,1,2,3,4*,5,6+,7,8**,9*+**
11112 0,1,2,3,4*,5,6+,7,8*,9**+**
11112 0,1,2,3,4*,5,6+,7,8,9***+**
11112 0,1,2,3,4+++,5^,6,7-,8*-,9/
11112 0,1,2,3,4^,5,6/-*+,7-,8*,9*
11112 0,1,2,3,4^,5,6/-*+,7-,8,9**
11112 0,1,2,3,4^,5,6/-*,7-+,8*,9*
11112 0,1,2,3,4^,5,6/-*,7-+,8,9**
11112 0,1,2,3,4,5+,6,7,89*-*-**
11112 0,1,2,34,5,67*,8-*++,9-
11112 0,1,2,34,5,67*,8-*+,9-+
11112 0,1,2,34,5,67*,8-*,9-++
11112 0,1,23+,4,5,6*,7,8+*+,9+*
11112 0,1,23+,4,5,6*,7,8+*,9++*
11112 0,1,23+,4,5,6,7,8+**+,9+*
11112 0,1,23+,4,5,6,7,8+**,9++*
11112 0,1,23+,456,7+,8,9-^/
11112 0,1,23+,456,7,8,9-*-*
11112 0,1,23+,456,7,8,9-/-*
11112 0,1,23,456*+,7,89*+
11112 0,1,23,456*,7,89*++
11112 0,12,3^,456,7+*,8/,9/
11112 0,12,3^,456,7+*,8,9*/
11112 0,12,3^,456,7+,8/*,9/
11112 0,12,3^,456,7+,8/,9/*
11112 0,12,3^,456,7+,8,9*/*
11112 0,12,3,4^,56+,789+*
11112 0,12,3,4^,56,789++*
11112 0,12,34,5*,6,78+,9*+*
11112 0,123,4,5,6/,7+*+,8*,9*
11112 0,123,4,5,6/,7+*+,8,9**

To take one example to convert to algebraic notation:

,1,2+,3,456,7+,8*+*,9-    is  (1+2) * (3 + (456+7)*8) - 9  = 11112

And Downward:

11112 0,9,8*,7*,6,5+*,4,3*+,2*,1*
11112 0,9,8*,7*,6,5+*,4,3*+,2*,1/
11112 0,9,8*,7*,6,5+*,4,3*+,2*,1^
11112 0,9,8*,7*,6,5+*,4,3*+,2,1**
11112 0,9,8*,7*,6,5+*,4,3*+,2,1/*
11112 0,9,8*,7*,6,5+*,4,3*+,2,1^*
11112 0,9,8*,7,6,5+**,4,3*+,2*,1*
11112 0,9,8*,7,6,5+**,4,3*+,2*,1/
11112 0,9,8*,7,6,5+**,4,3*+,2*,1^
11112 0,9,8*,7,6,5+**,4,3*+,2,1**
11112 0,9,8*,7,6,5+**,4,3*+,2,1/*
11112 0,9,8*,7,6,5+**,4,3*+,2,1^*
11112 0,9,8-,7-,6,5,43,2^+,1-*+
11112 0,9,8-,7-,6,5,43,2^,1-+*+
11112 0,9,8-,7,6,5,43,2^+,1-*--
11112 0,9,8-,7,6,5,43,2^,1-+*--
11112 0,9,8+,7+,6,5*,432+,1+*
11112 0,9,8+,7+,6,5*,432,1++*
11112 0,9,8+,7,6-,5+,43,2^*+,1+
11112 0,9,8+,7,6-,5+,43,2^*,1++
11112 0,9,8+,7,6,5--,43,2^*+,1+
11112 0,9,8+,7,6,5--,43,2^*,1++
11112 0,9,8+,7,6,5+,4,3*,2^*,1+*+
11112 0,9,8,7**,6,5+*,4,3*+,2*,1*
11112 0,9,8,7**,6,5+*,4,3*+,2*,1/
11112 0,9,8,7**,6,5+*,4,3*+,2*,1^
11112 0,9,8,7**,6,5+*,4,3*+,2,1**
11112 0,9,8,7**,6,5+*,4,3*+,2,1/*
11112 0,9,8,7**,6,5+*,4,3*+,2,1^*
11112 0,9,8,7*-,6,5,4^*+,3*,2+,1+
11112 0,9,8,7*-,6,5,4^*+,3*,2,1++
11112 0,9,8,7*,6+,5*,4*,3-*,21-
11112 0,9,8,7*,6+,5,4**,3-*,21-
11112 0,9,8,7*,6,5+**,4,3*+,2*,1*
11112 0,9,8,7*,6,5+**,4,3*+,2*,1/
11112 0,9,8,7*,6,5+**,4,3*+,2*,1^
11112 0,9,8,7*,6,5+**,4,3*+,2,1**
11112 0,9,8,7*,6,5+**,4,3*+,2,1/*
11112 0,9,8,7*,6,5+**,4,3*+,2,1^*
11112 0,9,8,7*,6,5+*,4,3/+*,2*,1*
11112 0,9,8,7*,6,5+*,4,3/+*,2*,1/
11112 0,9,8,7*,6,5+*,4,3/+*,2*,1^
11112 0,9,8,7*,6,5+*,4,3/+*,2,1**
11112 0,9,8,7*,6,5+*,4,3/+*,2,1/*
11112 0,9,8,7*,6,5+*,4,3/+*,2,1^*
11112 0,9,8,7*,6,5+*,4,3/+,2**,1*
11112 0,9,8,7*,6,5+*,4,3/+,2**,1/
11112 0,9,8,7*,6,5+*,4,3/+,2**,1^
11112 0,9,8,7*,6,5+*,4,3/+,2*,1**
11112 0,9,8,7*,6,5+*,4,3/+,2*,1/*
11112 0,9,8,7*,6,5+*,4,3/+,2*,1^*
11112 0,9,8,7*,6,5+*,4,3/+,2,1***
11112 0,9,8,7*,6,5+*,4,3/+,2,1/**
11112 0,9,8,7*,6,5+*,4,3/+,2,1^**
11112 0,9,8,7*,6,5,4^*--,3*,2+,1+
11112 0,9,8,7*,6,5,4^*--,3*,2,1++
11112 0,9,8,7*,6,5,4^*-,3*-,21+
11112 0,9,8,7*,6,5,4^*-,3*,21--
11112 0,9,8,7*,65*,4,3^+*,2,1+/
11112 0,9,8,7*,65*,4,3^+,2,1+/*
11112 0,9,8,7+-,6,5,43,2^+,1-*+
11112 0,9,8,7+-,6,5,43,2^,1-+*+
11112 0,9,8,7++,6,5*,432+,1+*
11112 0,9,8,7++,6,5*,432,1++*
11112 0,9,8,7+,6*,5*+,4+,3,21+*
11112 0,9,8,7+,6*,5*,4++,3,21+*
11112 0,9,8,7+,6,5**+,4+,3,21+*
11112 0,9,8,7+,6,5**,4++,3,21+*
11112 0,9,8,7+,6,5,43,2^+,1-*--
11112 0,9,8,7+,6,5,43,2^,1-+*--
11112 0,9,8,7,6-,5+,43,2^*++,1+
11112 0,9,8,7,6-,5+,43,2^*+,1++
11112 0,9,8,7,6-,5+,43,2^*,1+++
11112 0,9,8,7,6,5--,43,2^*++,1+
11112 0,9,8,7,6,5--,43,2^*+,1++
11112 0,9,8,7,6,5--,43,2^*,1+++
11112 0,9,8,7,6,5+***,4,3*+,2*,1*
11112 0,9,8,7,6,5+***,4,3*+,2*,1/
11112 0,9,8,7,6,5+***,4,3*+,2*,1^
11112 0,9,8,7,6,5+***,4,3*+,2,1**
11112 0,9,8,7,6,5+***,4,3*+,2,1/*
11112 0,9,8,7,6,5+***,4,3*+,2,1^*
11112 0,9,8,7,6,5+**,4,3/+*,2*,1*
11112 0,9,8,7,6,5+**,4,3/+*,2*,1/
11112 0,9,8,7,6,5+**,4,3/+*,2*,1^
11112 0,9,8,7,6,5+**,4,3/+*,2,1**
11112 0,9,8,7,6,5+**,4,3/+*,2,1/*
11112 0,9,8,7,6,5+**,4,3/+*,2,1^*
11112 0,9,8,7,6,5+**,4,3/+,2**,1*
11112 0,9,8,7,6,5+**,4,3/+,2**,1/
11112 0,9,8,7,6,5+**,4,3/+,2**,1^
11112 0,9,8,7,6,5+**,4,3/+,2*,1**
11112 0,9,8,7,6,5+**,4,3/+,2*,1/*
11112 0,9,8,7,6,5+**,4,3/+,2*,1^*
11112 0,9,8,7,6,5+**,4,3/+,2,1***
11112 0,9,8,7,6,5+**,4,3/+,2,1/**
11112 0,9,8,7,6,5+**,4,3/+,2,1^**
11112 0,9,8,7,6,5+,4,3*,2^*,1+*++
11112 0,9,8,7,6,5,4^*-,3/-,21-*
11112 0,9,8,7,6,5,4^*-,3/,21+-*
11112 0,9,8,7,6,5,43,2^+,1-*-+-
11112 0,9,8,7,6,5,43,2^,1-+*-+-
11112 0,9,8,7,65**,4,3^+*,2,1+/
11112 0,9,8,7,65**,4,3^+,2,1+/*
11112 0,9,8,7,65*+*,4*,3/,2*,1*
11112 0,9,8,7,65*+*,4*,3/,2*,1/
11112 0,9,8,7,65*+*,4*,3/,2*,1^
11112 0,9,8,7,65*+*,4*,3/,2,1**
11112 0,9,8,7,65*+*,4*,3/,2,1/*
11112 0,9,8,7,65*+*,4*,3/,2,1^*
11112 0,9,8,7,65*+*,4*,3,2//,1*
11112 0,9,8,7,65*+*,4*,3,2//,1/
11112 0,9,8,7,65*+*,4*,3,2//,1^
11112 0,9,8,7,65*+*,4*,3,2/,1*/
11112 0,9,8,7,65*+*,4*,3,2/,1//
11112 0,9,8,7,65*+*,4*,3,2/,1^/
11112 0,9,8,7,65*+*,4*,3,2,1*//
11112 0,9,8,7,65*+*,4*,3,2,1///
11112 0,9,8,7,65*+*,4*,3,2,1^//
11112 0,9,8,7,65*+*,4,3/*,2*,1*
11112 0,9,8,7,65*+*,4,3/*,2*,1/
11112 0,9,8,7,65*+*,4,3/*,2*,1^
11112 0,9,8,7,65*+*,4,3/*,2,1**
11112 0,9,8,7,65*+*,4,3/*,2,1/*
11112 0,9,8,7,65*+*,4,3/*,2,1^*
11112 0,9,8,7,65*+*,4,3/,2**,1*
11112 0,9,8,7,65*+*,4,3/,2**,1/
11112 0,9,8,7,65*+*,4,3/,2**,1^
11112 0,9,8,7,65*+*,4,3/,2*,1**
11112 0,9,8,7,65*+*,4,3/,2*,1/*
11112 0,9,8,7,65*+*,4,3/,2*,1^*
11112 0,9,8,7,65*+*,4,3/,2,1***
11112 0,9,8,7,65*+*,4,3/,2,1/**
11112 0,9,8,7,65*+*,4,3/,2,1^**
11112 0,9,8,7,65*+*,4,3,2//*,1*
11112 0,9,8,7,65*+*,4,3,2//*,1/
11112 0,9,8,7,65*+*,4,3,2//*,1^
11112 0,9,8,7,65*+*,4,3,2//,1**
11112 0,9,8,7,65*+*,4,3,2//,1/*
11112 0,9,8,7,65*+*,4,3,2//,1^*
11112 0,9,8,7,65*+*,4,3,2/,1*/*
11112 0,9,8,7,65*+*,4,3,2/,1//*
11112 0,9,8,7,65*+*,4,3,2/,1^/*
11112 0,9,8,7,65*+*,4,3,2,1*//*
11112 0,9,8,7,65*+*,4,3,2,1///*
11112 0,9,8,7,65*+*,4,3,2,1^//*
11112 0,9,8,7,65*+,4**,3/,2*,1*
11112 0,9,8,7,65*+,4**,3/,2*,1/
11112 0,9,8,7,65*+,4**,3/,2*,1^
11112 0,9,8,7,65*+,4**,3/,2,1**
11112 0,9,8,7,65*+,4**,3/,2,1/*
11112 0,9,8,7,65*+,4**,3/,2,1^*
11112 0,9,8,7,65*+,4**,3,2//,1*
11112 0,9,8,7,65*+,4**,3,2//,1/
11112 0,9,8,7,65*+,4**,3,2//,1^
11112 0,9,8,7,65*+,4**,3,2/,1*/
11112 0,9,8,7,65*+,4**,3,2/,1//
11112 0,9,8,7,65*+,4**,3,2/,1^/
11112 0,9,8,7,65*+,4**,3,2,1*//
11112 0,9,8,7,65*+,4**,3,2,1///
11112 0,9,8,7,65*+,4**,3,2,1^//
11112 0,9,8,7,65*+,4*,3/*,2*,1*
11112 0,9,8,7,65*+,4*,3/*,2*,1/
11112 0,9,8,7,65*+,4*,3/*,2*,1^
11112 0,9,8,7,65*+,4*,3/*,2,1**
11112 0,9,8,7,65*+,4*,3/*,2,1/*
11112 0,9,8,7,65*+,4*,3/*,2,1^*
11112 0,9,8,7,65*+,4*,3/,2**,1*
11112 0,9,8,7,65*+,4*,3/,2**,1/
11112 0,9,8,7,65*+,4*,3/,2**,1^
11112 0,9,8,7,65*+,4*,3/,2*,1**
11112 0,9,8,7,65*+,4*,3/,2*,1/*
11112 0,9,8,7,65*+,4*,3/,2*,1^*
11112 0,9,8,7,65*+,4*,3/,2,1***
11112 0,9,8,7,65*+,4*,3/,2,1/**
11112 0,9,8,7,65*+,4*,3/,2,1^**
11112 0,9,8,7,65*+,4*,3,2//*,1*
11112 0,9,8,7,65*+,4*,3,2//*,1/
11112 0,9,8,7,65*+,4*,3,2//*,1^
11112 0,9,8,7,65*+,4*,3,2//,1**
11112 0,9,8,7,65*+,4*,3,2//,1/*
11112 0,9,8,7,65*+,4*,3,2//,1^*
11112 0,9,8,7,65*+,4*,3,2/,1*/*
11112 0,9,8,7,65*+,4*,3,2/,1//*
11112 0,9,8,7,65*+,4*,3,2/,1^/*
11112 0,9,8,7,65*+,4*,3,2,1*//*
11112 0,9,8,7,65*+,4*,3,2,1///*
11112 0,9,8,7,65*+,4*,3,2,1^//*
11112 0,9,8,76*+,5,4+*,3+,2*,1*
11112 0,9,8,76*+,5,4+*,3+,2*,1/
11112 0,9,8,76*+,5,4+*,3+,2*,1^
11112 0,9,8,76*+,5,4+*,3+,2,1**
11112 0,9,8,76*+,5,4+*,3+,2,1/*
11112 0,9,8,76*+,5,4+*,3+,2,1^*
11112 0,9,8,76*,5+,4+*,3+,2*,1*
11112 0,9,8,76*,5+,4+*,3+,2*,1/
11112 0,9,8,76*,5+,4+*,3+,2*,1^
11112 0,9,8,76*,5+,4+*,3+,2,1**
11112 0,9,8,76*,5+,4+*,3+,2,1/*
11112 0,9,8,76*,5+,4+*,3+,2,1^*
11112 0,9,8,76*,5,4++*,3+,2*,1*
11112 0,9,8,76*,5,4++*,3+,2*,1/
11112 0,9,8,76*,5,4++*,3+,2*,1^
11112 0,9,8,76*,5,4++*,3+,2,1**
11112 0,9,8,76*,5,4++*,3+,2,1/*
11112 0,9,8,76*,5,4++*,3+,2,1^*
11112 0,9,87,65,4*+,32*+,1-
11112 0,9,87,65,4*+,32*,1-+
11112 0,987,6,5+,4+,3^,2,1+*+
11112 0,987,6,5,4++,3^,2,1+*+

Taking one example:

,9,87,65,4*+,32*,1-+  is  9 + (87 + 65*4)*32 - 1 = 11112

  Posted by Charlie on 2017-05-11 22:42:29
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