Ponder This Challenge - December 2022 - Unique Solution Letter Boxed

This puzzle was suggested by Victor Chang.

The New York Times' "Letter Boxed" puzzle consists of a square with three letters on each side - a total of 12 unique letters. In the puzzle, you write down a sequence of words while following these rules:

1. All words must be accepted words in English.
2. No two consecutive letters in a word can be from the same side of the square.
3. The last letter of a given word becomes the first letter of the subsequent word.
4. All the letters in the square must appear in the sequence of words in the solution.

For example, if the box consists of the following letters, three on each side: 'hou wcr tka nbl', a valid solution would be ['burnt', 'throw', 'warlock']. There is also a two-word solution, ['unclot', 'throwback'] if one accepts esoteric words (which is not necessarily the case for the NYT).

We are interested in finding puzzles with unique two-word solutions, even when the allowed dictionary is large. For this puzzle, we used the word list located here

Your goal: Find a box that has only a single two-word solution. Submit the box as a string, e.g., 'hou wcr tka nbl', with the two-word solution as well.

A bonus "*" will be given to those who find a sequence of at least 15 letters that also has a unique solution. This can be achieved by either adding more sides to the polygon (e.g., a pentagon with three letters on each side instead of a square) or by adding more letters to each side (e.g., 4 letters on each side of the square). Submit your solution in the same format as shown above.

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.

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Solution

• Click here to view the solution

  December 2022 Solution:

  David F.H. Dunkley have sent an elaborate explanation of his solution, given below – thanks David!

  12 letters -
  avd lbw peo htf
  alphabet , twelvefold

  15 letters (pentagon with three letters per side)
  ugy ner jlb dkw ios
  unjudgelike , eyebrows

  22 letters (hendecagon with two letters per side)
  cb ai nt ew ok rg fm lh ds yp ux
  cabinetworking , formaldehydesulphoxylic

  My program makes 26-bit masks of the letters from each word, and then compares all mask pairs with bitwise or operation to find the pairs of masks that combine to a "pair mask" with 12 bits set. Then it searches through each of the words that belong to each of those pairs of masks, to find how many have start/end letters in common.
  If there is only one, then that's a great candidate to build a letter box. It's fairly trivial to build the Letter Box, given two words that I know are unique in covering the 12 letters. If there is more than one word pair for the 12 letters, it would surely be possible to build a Letter Box that only allows one pair to work. I didn't follow that idea through.

  The program outputs a list of all the unique word pair candidates. There are 250,170 candidate unique word pairs, for twelve letters. I tried to pick relatively "ordinary" words for my solutions, as far as possible, because even though I am a native English speaker, most of the words in the list are really strange to me!

  It was simple to change the program to work with 15 letters or more. 22 letters was the most it could find a unique pair for.

Solvers

• *Lorenz Reichel (5/12/2022 11:19 AM IDT)
• Paul Revenant (5/12/2022 12:06 PM IDT)
• Lazar Ilic (5/12/2022 1:17 PM IDT)
• Laurence Warne (5/12/2022 4:13 PM IDT)
• *Giorgos Kalogeropoulos (5/12/2022 9:18 PM IDT)
• *Bert Dobbelaere (5/12/2022 10:28 PM IDT)
• Reiner Martin (5/12/2022 11:22 PM IDT)
• *Arthur Vause (6/12/2022 12:16 AM IDT)
• *Sanandan Swaminathan (6/12/2022 9:23 AM IDT)
• *Sri Mallikarjun J (6/12/2022 2:01 PM IDT)
• *Latchezar Christov (6/12/2022 2:02 PM IDT)
• Daniel Chong Jyh Tar (6/12/2022 2:33 PM IDT)
• Prashant Wankhede (6/12/2022 3:56 PM IDT)
• Lorenzo Gianferrari Pini (6/12/2022 6:55 PM IDT)
• *Michael Branicky (6/12/2022 7:20 PM IDT)
• Gary M. Gerken (7/12/2022 6:13 AM IDT)
• *Stéphane Higueret (7/12/2022 2:36 PM IDT)
• *Harald Bögeholz (7/12/2022 4:46 PM IDT)
• *Franciraldo Cavalcante (7/12/2022 9:17 PM IDT)
• Andreas Puccio (8/12/2022 11:53 AM IDT)
• Tamir Ganor & Shouky Dan (8/12/2022 4:17 PM IDT)
• *Amos Guler (8/12/2022 7:03 PM IDT)
• *Kennedy (8/12/2022 8:53 PM IDT)
• *Tim Walters (8/12/2022 10:09 PM IDT)
• *Chintan Maggu (9/12/2022 1:44 PM IDT)
• *Dieter Beckerle (9/12/2022 3:27 PM IDT)
• *Chris Shannon (9/12/2022 9:23 PM IDT)
• *Alper Halbutogullari (11/12/2022 10:00 PM IDT)
• *Sean Egan (12/12/2022 6:10 PM IDT)
• *Asaf Zimmerman (14/12/2022 12:09 AM IDT)
• *Nyles Heise (14/12/2022 4:56 AM IDT)
• Clay Pearson (14/12/2022 8:15 PM IDT)
• *David Greer (15/12/2022 12:30 PM IDT)
• *Evert van Dijken (15/12/2022 2:48 PM IDT)
• Stefan Wirth (15/12/2022 5:22 PM IDT)
• Marco Bellocchi (15/12/2022 10:38 PM IDT)
• *Andy Greig (17/12/2022 12:04 AM IDT)
• *David F.H. Dunkley (17/12/2022 12:34 AM IDT)
• Wilder Boyden (17/12/2022 5:39 AM IDT)
• *Peter Moser (18/12/2022 4:06 PM IDT)
• *Dominik Reichl (18/12/2022 6:47 PM IDT)
• *Tyler Mullen (19/12/2022 6:04 AM IDT)
• Thomas Mangan (19/12/2022 4:11 PM IDT)
• Todd Will (19/12/2022 11:47 PM IDT)
• *Martin Bisson (20/12/2022 11:28 AM IDT)
• *Vladimir Volevich (20/12/2022 6:08 PM IDT)
• *Guillaume Escamocher (21/12/2022 6:54 PM IDT)
• Shirish Chinchalkar (22/12/2022 3:21 AM IDT)
• *Hakan Summakoğlu (22/12/2022 4:11 PM IDT)
• *Daniel Bitin (23/12/2022 4:50 PM IDT)
• *Phil Proudman (23/12/2022 5:23 PM IDT)
• *Li Li (23/12/2022 7:53 PM IDT)
• *Kang Jin Cho (28/12/2022 9:09 PM IDT)
• Jeff Harris (29/12/2022 9:19 PM IDT)
• Justin Stanley (29/12/2022 10:29 PM IDT)
• Fabio Michele Negroni (30/12/2022 5:04 PM IDT)
• Sachal Mahajan (31/12/2022 6:35 AM IDT)
• Christoph Baumgarten (2/1/2023 10:38 AM IDT)
• *Bertram Felgenhauer (2/1/2023 12:58 PM IDT)
• *Motty Porat (4/1/2023 12:36 AM IDT)
• Radu-Alexandru Todor (7/1/2023 10:58 PM IDT)