# Crazy Representations of Natural Numbers – The 10958 Problem

In 2013-2014, author (click here) wrote natural numbers in increasing and decreasing orders of 1 to 9 using only basic operations. See below some examples:

This work has been considered as “Improbable Research”. For details see the links:

Also refer Nebus 1, Nebus 2  for more comments. We observe that the number 10958 is the only number among 0 to 11111 that is not possible to write in terms of 1 to 9 in the increasing case. We need to use extra operations such as square-root and/or factorial to write it. See below:

### 10958:=1+2+3!!+(-4+5!+6-7)*89.

Numberphile produced two videos on YouTube using concatenation as an extra operation to bring representation of number 10958 in increasing order. See below both the links:

(i) The 10,958 Problem – Numberphile – M. Parker; (1.968.177 views – till  09.03.19)
(ii)  A 10,958 Solution – Numberphile – M. Parker. (1.118.495 views- till 09.03.19)

RUSSIAN TRANSLATIONS

(i) The 10,958 Problem – Numberphile – M. Parker; (2.237.434 views – till 09.03.19)
(ii)  A 10,958 Solution – Numberphile – M. Parker. (369.036 views- till 09.03.19)

Total views: 5.693.142 till 09.03.19

#### Extensions

1. Extension to 11112-20000 – https://doi.org/10.5281/zenodo.2543626;
2. Extension to 20001-25000 – Taneja-2 ;
3. Extension to 25001-30000 – Taneja-3

FURTHER EXTENSIONS

For the results among 11112 to 30000 see the following link:   https://zenodo.org/record/1998118

More work from Anne E. Bras and V.H.J. van der Velden see the following links:
– https://zenodo.org/record/2276623
– https://zenodo.org/record/2525823
– https://zenodo.org/record/2530025

More work from Anne E. Bras and V.H.J. van der Velden in different bases:

– https://zenodo.org/record/2530974 (= base 11, 0 up to AAAA, incomplete)
– https://zenodo.org/record/2530976 (= base 12, 0 up to BBBB, incomplete)
– https://zenodo.org/record/2530978 (= base 13, 0 up to CCCC, incomplete)
– https://zenodo.org/record/2530980 (= base 14, 0 up to DDDD, incomplete)
– https://zenodo.org/record/2530982 (= base 15, 0 up to EEEE, incomplete)
– https://zenodo.org/record/2530984 (= base 16, 0 up to FFFF, complete)
– https://zenodo.org/record/2536047 (= base 16, -FFFFF up to FFFFF, complete)
– https://zenodo.org/record/2531777 (= base 11 up to 15, now all completed)

## 56 thoughts on “Crazy Representations of Natural Numbers – The 10958 Problem”

1. Max Blechman says:

Hello Mr. Taneja.

I was recently on YouTube looking at a video on the 10958 problem when I found a solution by the account Kostya Bardock. All credit for this solution belongs to them.

(1234-5)*6+7*(8^√9)

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