Ponder This Challenge - January 2026 - Number splitting

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 A₁₂₃ = {15, 24, 123} .

For a general natural number 𝑛 , we define Aₙ similarly as the set of all natural numbers that can be obtained by splitting and adding the decimal notation of 𝑛 . For example, A₃₁₆₅₈ = {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 · A ≜ { n · 𝑥 | 𝑥 ∈ A } (where · is the usual multiplication of numbers).

Your goal: Find the sum of all the natural numbers 𝑥 such that there exists 1 ≤ 𝑛 ≤ 10⁶ for which 𝑥 ∈ 𝑛 · Aₙ and 𝑛 ∈ Aₓ .

A bonus "*" will be given for finding the **sum **of all the natural numbers x such that there exists 1 ≤ n ≤ 10⁷ for which 𝑥 ∈ 𝑛 · Aₙ and 𝑛 ∈ Aₓ .

Solvers

• *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)
• *Suphakorn Khamwongsa (15/1/2026 4:51 AM IDT)
• Amos Guler (15/1/2026 10:19 PM IDT)
• Natthawat Thuisuwan (16/1/2026 6:03 AM IDT)
• Ricardo Koller (16/1/2026 9:49 PM IDT)
• *Hongcheng Zhu (16/1/2026 10:15 PM IDT)
• *Fabio Filatrella (17/1/2026 1:03 PM IDT)
• *Kang Jin Cho (17/1/2026 4:30 PM IDT)
• *Hansraj Nahata & Govind Jujare (19/1/2026 7:42 PM IDT)
• Jean-Michel Torres (20/1/2026 2:31 PM IDT)
• Mathias Schenker (21/1/2026 9:59 AM IDT)
• *Isabelle Mirouze (21/1/2026 11:58 AM IDT)
• *Preetham Poluparthi (21/1/2026 6:11 PM IDT)
• *Motty Porat (23/1/2026 4:48 AM IDT)
• *Nyles Heise (25/1/2026 4:28 AM IDT)
• *Chris Shannon (26/1/2026 6:48 AM IDT)
• *Armin Krauß (26/1/2026 1:48 PM IDT)
• *Khaled Alam (28/1/2026 3:21 AM IDT)
• Piyush Agrawal (28/1/2026 10:12 AM IDT)
• *Sanandan Swaminathan (29/1/2026 12:30 AM IDT)