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:

(i) http://www.improbable.com/2013/02/12/lots-of-numbers-plain-and-almost-simple/;

(ii) http://www.improbable.com/2013/06/08/lots-more-numbers-deemed-crazy-sequential/

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; (**2.970.174** views – till 22.10.21)

(ii) A 10,958 Solution – Numberphile – M. Parker. (**1.552.751** views- till 22.10.21)

**RUSSIAN TRANSLATIONS**

(i) The 10,958 Problem – Numberphile – M. Parker; (**6.010.701 **views – till 22.10.21)

(ii) A 10,958 Solution – Numberphile – M. Parker. (**1.110.973** views- till 22.10.21)

Total views: **11.644.599** till 22.10.21

#### Also this number has become a question of discussion at various web-sites. See the links below:

- https://www.primepuzzles.net/puzzles/puzz_864.htm;
- https://www.futilitycloset.com/?s=10958;
- https://puzzling.stackexchange.com/questions/51129/the-10-958-problem;
- https://puzzling.stackexchange.com/questions/47923/rendering-the-number-10-958-with-the-string-1-2-3-4-5-6-7-8-9;
- https://www.reddit.com/r/mathematics/comments/6hooq2/the_10958_problem/;
- http://digg.com/video/10958-math-problem;
- http://wssrmnn.net/index.php/2017/04/20/the-10958-problem;
- https://twitter.com/aap03102/status/854319054173220864;
- https://inderjtaneja.com/2017/12/02/10958-problem-1-5-millions-views.
- https://github.com/nvalis/10958;
- https://www.youtube.com/results?search_query=10958;
- https://www.youtube.com/watch?v=yCMvekktTa0.

**Extensions**

- Extension to 11112-20000 – https://doi.org/10.5281/zenodo.2543626;
- Extension to 20001-25000 – Taneja-2 ;
- Extension to 25001-30000 – Taneja-3.
- Revised: 20001-3000: Crazy Representations of Natural Numbers From 20001 to 30000, Zenodo, February 3, 2019, pp. 1-276,
**http://doi.org/10.5281/zenodo.2556036**.

**FURTHER EXTENSIONS**

- By use of different operations,
**Bill Ames**(AMES1, AMES2), recently, extended the above representations up to 1,111,111 in ascending orders of 1 to 9. The first link AMES1 is YouTube vídeo, and the second link AMES2 is txt file. - Also extension to small bases refer:
**T. WYLIE**, Crazy Sequential Representations of Numbers for Small Bases. **Anne E. Bras**and**V.H.J. van der Velden**extended the results from – 2147483647 up to 214748364 using only the basic operations. As the numbers increases the gaps also increases. For more details refer the following links:- https://zenodo.org/record/1288822

– https://zenodo.org/record/1288892

– https://zenodo.org/record/1292115

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)

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