perplexus.info :: Numbers : Inder Taneja's discovery

Inder Taneja's discovery (Posted on 2017-05-11)

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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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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