January2026 - Challenge
The decimal notation string of the number 123 can be split in various ways: To 12 and 3, to 1 and 23, and to 123. For each way to split the number, we add up the obtained parts. For 12 and 3, we obtain the number 15; for 1 and 23, we obtain the number 24; and for 123, we obtain 123 itself. We denote this by A_{123}=\{15, 24, 123\}.
For a general natural number n, we define A_n similarly as the set of all natural numbers that can be obtained by splitting and adding the decimal notation of n. For example, A_{31658} = \{23, 32, 50, 68, 77, 95, 104, 176, 329, 374, 662, 689, 1661, 3173, 31658\}
Given a set A of integers and an integer n we use the standard notation n\cdot A\triangleq\{n\cdot x\ |\ x\in A\} (where \cdot is the usual multiplication of numbers).
Your goal: Find the sum of all the natural numbers x such that there exists 1\le n\le 10^{6} for which x\in n\cdot A_n and n\in A_x.
A bonus "*" will be given for finding the sum of all the natural numbers x such that there exists 1\le n\le 10^{7} for which x\in n\cdot A_n and n\in A_x.
We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!
We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.
If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com
Challenge:
30/12/2025 @ 10:10 AM EST
Solution:
05/02/2026 @ 12:00 AM EST
List Updated:
19/01/2026 @ 10:10 AM EST
*Alper Halbutogullari(30/12/2025 10:54 AM IDT)
*Nadir S.(30/12/2025 11:20 AM IDT)
*Lazar Ilic(30/12/2025 11:33 AM IDT)
Alex Fleischer(30/12/2025 12:57 PM IDT)
*Bertram Felgenhauer(30/12/2025 1:05 PM IDT)
*Ahmet Yüksel(30/12/2025 3:11 PM IDT)
*Ankit Aggarwal(30/12/2025 3:23 PM IDT)
Abraham AJ Arshad(30/12/2025 3:58 PM IDT)
*Paul Lupascu(30/12/2025 6:02 PM IDT)
*Dan Dima(30/12/2025 6:38 PM IDT)
*Prashant Wankhede(30/12/2025 9:08 PM IDT)
*George Jiri Spitalsky(30/12/2025 10:14 PM IDT)
*Reiner Martin(31/12/2025 12:19 AM IDT)
*Jean-François Hermant(31/12/2025 1:56 AM IDT)
Loukas Sidiropoulos(31/12/2025 2:14 AM IDT)
*Sergey Batalov(31/12/2025 2:58 AM IDT)
*King Pig(31/12/2025 3:37 AM IDT)
*Guangxi Liu(31/12/2025 10:29 AM IDT)
*Bert Dobbelaere(31/12/2025 12:42 PM IDT)
Michael Branicky(31/12/2025 11:17 PM IDT)
Gary M. Gerken(31/12/2025 11:57 PM IDT)
*Asaf Zimmerman(1/1/2026 7:04 PM IDT)
*Daniel Bitin(1/1/2026 7:05 PM IDT)
*Karl-Heinz Hofmann(1/1/2026 8:23 PM IDT)
*John Tromp(1/1/2026 8:57 PM IDT)
*Shirish Chinchalkar(2/1/2026 3:07 AM IDT)
*Robert Berec(2/1/2026 3:32 AM IDT)
*Stephen Ebert(2/1/2026 8:11 AM IDT)
*Latchezar Christov(2/1/2026 8:29 AM IDT)
*Oscar Volpatti(2/1/2026 10:55 AM IDT)
*Gil Citro(2/1/2026 6:25 PM IDT)
Paul Revenant(2/1/2026 9:25 PM IDT)
*Rodrigo Fresneda(2/1/2026 9:50 PM IDT)
Yosef Nago(2/1/2026 10:43 PM IDT)
*Juergen Koehl(3/1/2026 11:00 AM IDT)
*Liubing Yu(3/1/2026 11:26 AM IDT)
Lorenz Reichel(3/1/2026 1:37 PM IDT)
Hakan Summakoğlu(3/1/2026 6:52 PM IDT)
David Keyes(4/1/2026 1:03 AM IDT)
*Shouky Dan & Tamir Ganor(4/1/2026 9:48 AM IDT)
*Fakih Karademir(4/1/2026 3:40 PM IDT)
*Franciraldo Cavalcante(4/1/2026 4:32 PM IDT)
*Arthur Vause(4/1/2026 7:21 PM IDT)
*Dominik Reichl(4/1/2026 9:48 PM IDT)
*Michael Vahle(4/1/2026 10:31 PM IDT)
Giorgos Kalogeropoulos(4/1/2026 11:24 PM IDT)
*Dieter Beckerle(4/1/2026 11:45 PM IDT)
*Martin Thorne(5/1/2026 12:57 AM IDT)
Sherman Yuen(5/1/2026 7:06 AM IDT)
*Teerachot Saenthong(5/1/2026 9:20 AM IDT)
*Stéphane Higueret(5/1/2026 12:08 PM IDT)
*Vladimir Volevich(5/1/2026 3:13 PM IDT)
*Kandala Savitha Viswanadh(5/1/2026 6:22 PM IDT)
Rishikesh Pandey(5/1/2026 7:37 PM IDT)
*Peter Moser(5/1/2026 9:41 PM IDT)
*Florian Sikora(6/1/2026 12:08 AM IDT)
*Daniel Chong Jyh Tar(7/1/2026 5:50 PM IDT)
*Ankit Chakraborty(7/1/2026 11:00 PM IDT)
*Hugo Pfoertner(8/1/2026 3:26 PM IDT)
*Lorenzo Gianferrari Pini(8/1/2026 4:07 PM IDT)
Christoph Schmidt(8/1/2026 4:26 PM IDT)
Evan Semet(8/1/2026 7:06 PM IDT)
*Michael Liepelt(8/1/2026 9:27 PM IDT)
*Justin Mazenauer(9/1/2026 4:08 PM IDT)
*Todd Will(10/1/2026 8:50 PM IDT)
Ken Dsouza(11/1/2026 2:16 AM IDT)
*Alex Izvalov(11/1/2026 9:40 AM IDT)
*Daniel Lincke(11/1/2026 11:11 PM IDT)
*David F.H. Dunkley(14/1/2026 12:36 AM IDT)
Naftali Peles(14/1/2026 4:32 AM IDT)
*Jan Ondras(14/1/2026 10:24 AM IDT)