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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.

## September 2022 - Challenge

The IBM Research Lab in Israel has hosted and produced IBM’s Ponder This monthly challenge for the past fifteen years. Our lab is now changing its official name from IBM Research – Haifa (sometimes called the Haifa Research Lab, or HRL) to IBM Research - Israel (ISRL). To celebrate the name change, and the 50th anniversary of the lab we’re also marking this year, this month we challenge you to arrange the letters "ISRL" on a grid such that the result is non-overlapping and symmetrical. We say that an NxM grid G is "symmetrical" if either G[i][j] == G[i][M-1-j] for 0 <= i <= N-1 and 0 <= j <= M-1, or G[i][j] == G[N-1-i][j] for each such i,j.

For example, the following HRL letters

```H H RR  L
HHH RR  L
H H R R LLL
```

can be arranged into a symmetrical shape as follows (assuming all the grid points in the letters are essentially generic points):

```LLLRH H
LRR HHH
LRRRH H
```

Your goal: Find a way to write the letters ISRL in such a way such that a symmetrical shape can be constructed from all of them. The letters must be legible and as close as possible to the following shapes:

```I SSS RR  L
I S   R R L
I SSS RR  L
I   S R R L
I SSS R R LLL
```
1) The coordinates of the letters in ISRL order.
For example, the letters H, R, L can be written as grids in the following manner:

H: [[0,0],[0,1],[0,2],[1,1],[2,0],[2,1],[2,2]]
R: [[0,0],[0,1],[0,2],[1,1],[1,2],[2,0]]
L: [[0,0],[0,1],[0,2],[1,0],[2,0]]

2) A representation of the letters’ arrangement in a symmetrical shape, using a list of elements of the form [x,y,r], one element list per letter, where r is the number of 90-degree clockwise rotations of the shape, and (x,y) is the coordinate in the solution grid where the letter should be placed, such that the (0,0) coordinate of the letter *after being rotated* is placed there. In the rearranged HRL example above, the solution is written as [4,0,0], [1,0,3], [0,0,1]

So the format of the solution in this case is

[[0,0],[0,1],[0,2],[1,1],[2,0],[2,1],[2,2]]
[[0,0],[0,1],[0,2],[1,1],[1,2],[2,0]]
[[0,0],[0,1],[0,2],[1,0],[2,0]]
[[4,0,0], [1,0,3], [0,0,1]]

A Bonus "*" will be given for a symmetrical shape that uses all the letters H R L I S R L (the letters for "R" and "L" should be used twice and identical in both uses). Again, you have flexibility in choosing the exact shape of the letter.

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: 31/08/2022 @ 21:30:00 PM EST
Solution: 04/10/2022 @ 12:00 PM EST
List Updated: 30/10/2022 @ 21:30 PM EST

Lorenz Reichel (5/9/2022 11:35 AM IDT)
*Bertram Felgenhauer (5/9/2022 9:10 PM IDT)
*Alper Halbutogullari (6/9/2022 8:07 AM IDT)
Andreas Puccio (6/9/2022 3:32 PM IDT)
Daniel Chong Jyh Tar (7/9/2022 8:09 AM IDT)
John Tromp (7/9/2022 7:40 PM IDT)
Victor Chang (8/9/2022 5:44 AM IDT)
Vardhan Walavalkar (9/9/2022 7:25 AM IDT)
Guillaume Escamocher (10/9/2022 4:43 PM IDT)
*Dieter Beckerle (10/9/2022 10:53 PM IDT)
Gary M. Gerken (11/9/2022 8:06 PM IDT)
*John Spurgeon (12/9/2022 3:19 AM IDT)
Daniel Bitin (13/9/2022 2:21 PM IDT)
*David Greer (14/9/2022 9:14 PM IDT)
Amos Guler (14/9/2022 9:16 PM IDT)
*Lazar Ilic (15/9/2022 7:28 PM IDT)
Nyles Heise (19/9/2022 8:11 AM IDT)
Lorenzo Gianferrari Pini (19/9/2022 11:07 AM IDT)
*Motty Porat (25/9/2022 2:56 AM IDT)
Tim Walters (25/9/2022 3:59 PM IDT)
*Sanandan Swaminathan (27/9/2022 1:11 AM IDT)
Dan Dima (29/9/2022 6:22 PM IDT)
Fabio Michele Negroni (30/9/2022 12:21 PM IDT)
*Kang Jin Cho (30/9/2022 9:25 PM IDT)
Oscar Volpatti (1/10/2022 4:23 AM IDT)
Radu-Alexandru Todor (2/10/2022 2:58 PM IDT)
*Élie Goudout (3/10/2022 4:02 PM IDT)
Paul Revenant (3/10/2022 5:07 PM IDT)
Li Li (4/10/2022 8:05 AM IDT)
Stéphane Higueret (5/10/2022 8:26 PM IDT)
Phil Proudman (6/10/2022 12:46 AM IDT)