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# Ponder This

Welcome to our monthly puzzles.
You are cordially invited to match wits with some of the best minds in IBM Research.

## October 2022 - Challenge

The following puzzle was suggested by Evert van Dijken: Consider a n\times n\times n cube. It has 3n^2 rows, 6n diagonals from one edge to the opposite edge, and 4 corner-to-corner diagonals.

To each 1\times1\times 1 cube, we assign one of the numbers 1\dots n. The big cube does not need to have the same number of 1's, 2's.., or n's. Given such a numbering, we count the number of rows and diagonals that contain all the numbers 1\ldots n at once. As an example, consider for n=3 the numbering given by
1 3 2
2 1 3
3 2 1

2 1 3
3 2 1
1 3 2

3 2 1
1 3 2
2 1 3

There are 27 rows, each containing all the numbers 1, 2, and 3. To illustrate, we mark three such rows, each in a different axis, using different bracket styles:
[1] [3] [2]
2   1   3
3   2  {1}

(2)  1   3
(3)  2   1
(1)  3  {2}

3   2   1
1   3   2
2   1  {3}

There are 4 corner-to-corner diagonals, three of which contain 1, 2, 3 and one containing only 2:
[1]  3   {2}
2   1    3
3   2    1

2   1    3
3 {[2]}  1
1   3    2

3   2    1
1   3    2
{2}  1   [3]

And 18 edge diagonals, 9 of which contain 1, 2, and 3, while the other 9 do not:
[1] {3}   2
2   1    3
3   2    1

2  [1]   3
3  {2}   1
1   3    2

(3)   2   [1]
1  (3)    2
2  {1}   (3)

The total count is 27+9+3=39. We can compactly represent this solution by the string
1 3 2
2 1 3
3 2 1
2 1 3
3 2 1
1 3 2
3 2 1
1 3 2
2 1 3
39


(Note the count in the final line).

Your goal: Find a solution of at least 110 forn=6. Supply your solution in the above format.

A bonus "*" will be given for getting at least 60 solving the case n=5 when maximizing the count over diagonals (both edge and corner) first. A solution with more diagonals is always better, even if the final count is lower, but for the best diagonal count we still wish to maximize the total count. Supply your solution in the same format as above (no need to explicitly write down the diagonal count, only the total count).

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: 03/10/2022 @ 10:30:00 AM EST
Solution: 16/11/2022 @ 12:05 PM EST
List Updated: 16/11/2022 @ 12:05 PM EST

Paul Revenant (6/10/2022 12:48 PM IDT)
*Alper Halbutogullari (8/10/2022 12:50 AM IDT)
Alex Fleischer (8/10/2022 9:46 AM IDT)
Fabio Michele Negroni (8/10/2022 3:16 PM IDT)
*Bert Dobbelaere (9/10/2022 5:49 PM IDT)
*Evert van Dijken (10/10/2022 1:46 PM IDT)
Eran Vered (10/10/2022 5:21 PM IDT)
*Sanandan Swaminathan (10/10/2022 10:06 PM IDT)
Gary M. Gerken (11/10/2022 6:20 AM IDT)
Graham Hemsley (11/10/2022 6:33 PM IDT)
Daniel Bitin (11/10/2022 11:33 PM IDT)
Victor Chang (12/10/2022 5:56 AM IDT)
*Dieter Beckerle (12/10/2022 10:47 AM IDT)
*Lorenz Reichel (12/10/2022 4:22 PM IDT)
*Lazar Ilic (12/10/2022 7:28 PM IDT)
*Tim Walters (13/10/2022 3:27 AM IDT)
Stéphane Higueret (13/10/2022 9:41 AM IDT)
Jonathan Evans (13/10/2022 3:15 PM IDT)
Lorenzo Gianferrari Pini (13/10/2022 5:29 PM IDT)
*Frank Schoeps (13/10/2022 9:35 PM IDT)
*Bertram Felgenhauer (13/10/2022 11:32 PM IDT)
John Tromp (13/10/2022 11:50 PM IDT)
*Vladimir Volevich (14/10/2022 2:42 PM IDT)
*Jack Morris (15/10/2022 5:26 AM IDT)
*David Greer (15/10/2022 3:06 PM IDT)
*Dominik Reichl (15/10/2022 4:32 PM IDT)
Sri Mallikarjun J (15/10/2022 5:02 PM IDT)
Phil Proudman (16/10/2022 8:29 PM IDT)
Aditya Dalal (19/10/2022 6:18 AM IDT)
Shouky Dan & Tamir Ganor (19/10/2022 8:31 AM IDT)
Amos Guler (19/10/2022 10:53 AM IDT)
Nyles Heise (19/10/2022 7:38 PM IDT)
Andy Greig (20/10/2022 2:40 PM IDT)
Prashant Wankhede (21/10/2022 3:43 AM IDT)
*Latchezar Christov (21/10/2022 12:08 PM IDT)
*Li Li (23/10/2022 6:56 AM IDT)
Karl Mahlburg (24/10/2022 5:30 AM IDT)
Daniel Chong Jyh Tar (24/10/2022 3:45 PM IDT)
Dan Dima (24/10/2022 7:09 PM IDT)
Andreas Puccio (25/10/2022 12:52 AM IDT)
Kang Jin Cho (25/10/2022 8:08 PM IDT)
Maksym Voznyy (27/10/2022 11:33 PM IDT)
Jesus Vergara Temprado (28/10/2022 4:27 PM IDT)
*Chris Shannon (29/10/2022 1:15 AM IDT)
*Motty Porat (29/10/2022 3:37 AM IDT)
Light Bulb (29/10/2022 11:58 AM IDT)
Patrik Holopainen (29/10/2022 4:05 PM IDT)
Ali Gurel (30/10/2022 5:02 AM IDT)
Vardhan Walavalkar (30/10/2022 7:14 AM IDT)
Marco Bellocchi (31/10/2022 1:45 AM IDT)
David F.H. Dunkley (1/11/2022 12:20 AM IDT)
Oscar Volpatti (1/11/2022 3:48 AM IDT)
Harald Bögeholz (1/11/2022 1:39 PM IDT)
Radu-Alexandru Todor (4/11/2022 3:05 AM IDT)