Ponder This Challenge - January 2024 - A child's set of equations

This riddle was inspired by true events - thanks, true events!

A third-grade elementary school student received a riddle for the holidays:Fill in the 16 blank spaces on the board with the numbers $1, 2,\ldots,16$without repetitions (every number appears exactly once) such that all the equations are satisfied.

Here is a textual representation of the board, with the empty number cells denoted by '??':

A solution can be represented as a list with 16 elements, such that the first 4 elements are the values of the variables in the first row, the next 4 belong to the second row, etc.

Your goal: Find all solutions to the above problem. In your submission, please supply only one such solution (as a list, as described above) along with the total number of solutions. If you find an interesting way to solve the problem in a way that a third-grade student might come up with, we'd like to hear how you did it.

A bonus "*" will be given for finding two solutions of the above equations such that

1. The solutions differ in all places (i.e., there is no cell with the same number in both solutions)
2. It is possible to change at least half of the plus/minus signs in the board such that boths solutions still give the same results for every row and column.

Please note: When you change at least half the plus/minus signs, you can also change the resulting numbers. The results need not be the same as the original board, they only need to be the same for the two solutions. Give your submission by writing out the two requested solutions, followed by a list of the plus/minus signs given as a row-by-row list. The original board is represented by the following list where the first row is listed first, the second row second, etc.

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

Solution

• Click here to view the solution

  January 2024 Solution:

  There are 84 solutions in total to the original board. Of those, there are three pairs of solutions for which a solution to the bonus part can be given:

  [8, 13, 5, 11, 10, 14, 1, 15, 3, 12, 2, 16, 4, 7, 9, 6]
  [12, 8, 9, 6, 14, 10, 2, 16, 4, 7, 1, 11, 13, 3, 5, 15]
  [-,-,+,+,-,+,+,+,-,+,+,-,+,+,+,+,-,-,+,+,+,+,-,-]   
  
  [15,10,8,12,13,7,1,11,3,5,2,9,14,4,6,16]
  [16,12,9,14,10,6,2,8,4,7,1,11,13,3,5,15]
  [-,-,+,+,-,-,+,-,-,-,+,-,+,+,+,+,-,-,+,-,+,-,-,+]  
  
  [13,12,6,14,5,9,4,8,1,10,7,11,2,3,16,15]
  [14,8,2,15,4,16,1,11,5,9,3,10,6,7,13,12]
  [-,+,-,+,+,+,-,+,+,-,-,-,-,+,+,+,-,+,-,-,-,-,-,+]
  

  The straightforward method to solve the problem is using a constraint satisfaction solver. One can also attempt backtracking algorithms specifically suitable for the problem itself. However, not a single solver has managed to find a solution method suitable for a third-grade student, and it seems none exists.

Solvers

• Reiner Martin (31/12/2023 5:23 PM IDT)
• Daniel Johnson-Chyzhykov (31/12/2023 6:07 PM IDT)
• *Evert van Dijken (31/12/2023 6:36 PM IDT)
• *Harald Bögeholz (31/12/2023 6:55 PM IDT)
• Florian Sikora (31/12/2023 7:31 PM IDT)
• *Lazar Ilic (31/12/2023 7:33 PM IDT)
• *Sanandan Swaminathan (31/12/2023 8:25 PM IDT)
• *Ganor Tamir & Shouky Dan (31/12/2023 8:59 PM IDT)
• *Daniel Chong Jyh Tar (31/12/2023 9:19 PM IDT)
• Lorenz Reichel (31/12/2023 9:38 PM IDT)
• Evan Semet (31/12/2023 9:53 PM IDT)
• *Bertram Felgenhauer (31/12/2023 11:30 PM IDT)
• Kennedy (31/12/2023 11:44 PM IDT)
• *Yan-Wu He (31/12/2023 11:56 PM IDT)
• Ralf Jonas (1/1/2024 12:22 AM IDT)
• Shirish Chinchalkar (1/1/2024 1:05 AM IDT)
• *Fakih Karademir (1/1/2024 2:19 AM IDT)
• *Alper Halbutogullari (1/1/2024 3:34 AM IDT)
• *Gabe Stang & Zach (1/1/2024 3:38 AM IDT)
• Blaine Hill (1/1/2024 3:41 AM IDT)
• *Gary M. Gerken (1/1/2024 5:45 AM IDT)
• Dheeraj Reddy (1/1/2024 11:45 AM IDT)
• Richard Gosiorovsky (1/1/2024 11:56 AM IDT)
• *Yi Jiang (1/1/2024 1:34 PM IDT)
• *Amos Guler (1/1/2024 2:37 PM IDT)
• Hugo Pfoertner (1/1/2024 3:17 PM IDT)
• *Bert Dobbelaere (1/1/2024 3:28 PM IDT)
• Noam Zaks (1/1/2024 3:28 PM IDT)
• *Clive Tong (1/1/2024 6:38 PM IDT)
• *Guillaume Escamocher (1/1/2024 7:18 PM IDT)
• *Victor Chang (1/1/2024 7:22 PM IDT)
• *Hakan Summakoğlu (1/1/2024 7:33 PM IDT)
• Alex Fleischer (1/1/2024 7:34 PM IDT)
• *Andrew Gauld (1/1/2024 8:53 PM IDT)
• *Nadav Rubinstein (1/1/2024 9:22 PM IDT)
• *Michael Liepelt (1/1/2024 9:25 PM IDT)
• *Stéphane Higueret (1/1/2024 10:18 PM IDT)
• *Todd Will (1/1/2024 10:20 PM IDT)
• *Dieter Beckerle (1/1/2024 11:06 PM IDT)
• *Michał Franczak (1/1/2024 11:38 PM IDT)
• *Dominik Reichl (2/1/2024 12:10 AM IDT)
• *John Spurgeon (2/1/2024 1:46 AM IDT)
• Marcelo De Barros (2/1/2024 6:46 AM IDT)
• Fabio Michele Negroni (2/1/2024 1:16 PM IDT)
• *Guglielmo Sanchini (2/1/2024 1:52 PM IDT)
• *Asaf Zimmerman (2/1/2024 3:29 PM IDT)
• *Lyes Belhoul (2/1/2024 4:37 PM IDT)
• Michael Branicky (2/1/2024 7:16 PM IDT)
• *Franciraldo Cavalcante (2/1/2024 7:59 PM IDT)
• Andrew Scherbakov (3/1/2024 12:25 AM IDT)
• Eric Manalac (3/1/2024 11:11 AM IDT)
• Andreas Knüpfer (3/1/2024 3:38 PM IDT)
• Paul Revenant (3/1/2024 5:36 PM IDT)
• *Serino Dario (3/1/2024 6:57 PM IDT)
• *Vladimir Volevich (3/1/2024 9:48 PM IDT)
• *Tim Walters (3/1/2024 10:43 PM IDT)
• *Alain Michiels (3/1/2024 11:44 PM IDT)
• *Daniel Bitin (3/1/2024 11:52 PM IDT)
• *Loukas Sidiropoulos (4/1/2024 12:11 AM IDT)
• Ashfaque Shaikh (4/1/2024 7:17 AM IDT)
• Adrian Neacsu (4/1/2024 5:04 PM IDT)
• Nikhil Mishra (4/1/2024 7:47 PM IDT)
• Christoph Baumgarten (4/1/2024 7:50 PM IDT)
• *Grant Boudreaux (4/1/2024 10:36 PM IDT)
• Stefan Wirth (5/1/2024 12:05 AM IDT)
• Thomas Egense (5/1/2024 12:38 AM IDT)
• Marcel Caria (5/1/2024 1:00 AM IDT)
• Maksym Voznyy (5/1/2024 1:29 AM IDT)
• Austin Lee (5/1/2024 3:27 AM IDT)
• *Cameron Ritson (5/1/2024 3:40 AM IDT)
• *Paul Shaw (5/1/2024 1:11 PM IDT)
• *Armin Krauss (5/1/2024 4:52 PM IDT)
• Reda Kebbaj (5/1/2024 10:22 PM IDT)
• *Rong Fan (5/1/2024 10:38 PM IDT)
• Karl D'Souza (5/1/2024 11:50 PM IDT)
• *Gyozo Nagy (6/1/2024 12:56 AM IDT)
• *Aryan Soman (6/1/2024 1:31 AM IDT)
• Alexander Marschoun (6/1/2024 2:39 AM IDT)
• *Chris Shannon (6/1/2024 2:55 AM IDT)
• mmurphy12@nycap.rr.com (6/1/2024 6:13 AM IDT)
• Puvichakravarthy Ramachandran (6/1/2024 10:11 AM IDT)
• *Eden Saig (6/1/2024 12:45 PM IDT)
• *Latchezar Christov (6/1/2024 3:07 PM IDT)
• *Max Aehle (6/1/2024 3:44 PM IDT)
• *Paul Lupascu (6/1/2024 5:18 PM IDT)
• Albert Stadler (6/1/2024 5:38 PM IDT)
• *Ivan Lazaric (6/1/2024 7:59 PM IDT)
• Phil Proudman (6/1/2024 9:08 PM IDT)
• *Alex Izvalov (6/1/2024 10:37 PM IDT)
• Matt Cristina (7/1/2024 3:12 AM IDT)
• *Jiajun Li (7/1/2024 12:03 PM IDT)
• Arthur Vause (7/1/2024 2:47 PM IDT)
• *Thomas Mangan (7/1/2024 3:54 PM IDT)
• Hansraj Nahata (7/1/2024 6:08 PM IDT)
• Teawoo Kim (7/1/2024 6:17 PM IDT)
• *David F.H. Dunkley (7/1/2024 6:36 PM IDT)
• *Martin Thorne (7/1/2024 9:46 PM IDT)
• *Jingran Lin (8/1/2024 1:55 AM IDT)
• *Alexander Daryin (8/1/2024 1:47 PM IDT)
• *Zoltan Haindrich (8/1/2024 10:13 PM IDT)
• Giorgos Kalogeropoulos (9/1/2024 12:05 AM IDT)
• *Prashant Wankhede (9/1/2024 1:18 AM IDT)
• Naszvadi Peter (9/1/2024 1:31 PM IDT)
• *Sri Mallikarjun J (9/1/2024 7:18 PM IDT)
• Dan Dima (9/1/2024 8:35 PM IDT)
• Alireza Soroudi (10/1/2024 1:46 AM IDT)
• Lavanya Kannan (10/1/2024 6:59 AM IDT)
• *Lawrence Au (10/1/2024 2:38 PM IDT)
• Arvind Venkatesan (10/1/2024 7:11 PM IDT)
• *Marco Bellocchi (11/1/2024 1:44 AM IDT)
• Hriday Ajinkya (11/1/2024 5:30 AM IDT)
• Alok Menghrajani (12/1/2024 1:00 PM IDT)
• Kang Jin Cho (12/1/2024 6:36 PM IDT)
• Hao Wu (13/1/2024 10:29 PM IDT)
• *Dezhi Zhao (14/1/2024 6:13 AM IDT)
• *Karl Mahlburg (16/1/2024 4:23 AM IDT)
• Joaquim Neves Carrapa (16/1/2024 11:36 AM IDT)
• *Hamidreza Bidar (16/1/2024 7:10 PM IDT)
• Vamsi Krishna Talupula (17/1/2024 7:28 PM IDT)
• *José Soares (18/1/2024 4:56 PM IDT)
• Julius Barth (19/1/2024 8:35 AM IDT)
• Guang-Zhong Sun (19/1/2024 10:02 AM IDT)
• *Fabio Filatrella (19/1/2024 3:17 PM IDT)
• *Sofia Kozlova (21/1/2024 3:05 AM IDT)
• *Mark Pervovskiy (21/1/2024 12:03 PM IDT)
• Ulf Brännlund (21/1/2024 7:04 PM IDT)
• *Davide Colì (21/1/2024 10:46 PM IDT)
• *Julien Vincent Pradier (21/1/2024 10:52 PM IDT)
• *David Greer (22/1/2024 1:45 PM IDT)
• Rudy Cortembert (24/1/2024 11:14 PM IDT)
• *Nyles Heise (25/1/2024 5:16 AM IDT)
• *Li Li (25/1/2024 9:06 AM IDT)
• Pavel Stoimenov (25/1/2024 7:57 PM IDT)
• Greg Janée (27/1/2024 10:50 PM IDT)
• *Niko Kaso (28/1/2024 4:11 AM IDT)
• *Kevin Bauer (28/1/2024 3:46 PM IDT)
• *Dario Filatrella (28/1/2024 9:50 PM IDT)
• *Arnold Yim (29/1/2024 8:37 PM IDT)
• *Wenjie Yang (30/1/2024 6:25 AM IDT)
• Daniel Lincke (30/1/2024 6:59 PM IDT)
• *Motty Porat (31/1/2024 4:02 PM IDT)
• Elena Pavlova (31/1/2024 7:59 PM IDT)
• *Andrea Andenna (31/1/2024 8:17 PM IDT)
• Joe Hypoman (31/1/2024 10:59 PM IDT)
• *Oscar Volpatti (1/2/2024 1:02 AM IDT)
• *Vaskor Basak (3/2/2024 1:50 AM IDT)
• Radu-Alexandru Todor (3/2/2024 4:39 PM IDT)
• *Sampath Baddam (3/2/2024 10:19 PM IDT)
• Turgay Yuktas (4/2/2024 9:58 PM IDT)
• Sefi Ladkani (5/2/2024 12:51 AM IDT)