The New Sudoku Players' Forum

[Mathimagics]

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WORDLE, of course, refers to Josh Wardle's game (Wordle), but the observations below are probably applicable to the many clones that have appeared.

The standard elements are:

• DICT words: the dictionary, ie word list, the set of all 5 letter words that are allowed for any guess.

• GAME words: a subset (~20% of the DICT) of words that can be selected by the app as the code word (target word).

The questions that interest me are these:

• Q1: What are the best word(s) to use for the initial guess?

• Q2: Would a 6-letter WORDLE be harder?

To investigate these questions, I created a simple robotic player, which I call Marvin.

The approach is quite simple. Each game simulation involves fixing a target word, and one or two initial guess words, and then:

• tests the initial guess(es), obtaining a result code (Yellow, Green, Grey) (eg: [YXXGX])

• enters the guess loop:
  
  
  • from the result codes identify the implied constraints implied (letters present, letters excluded, etc)
  • find all words that satisfy these constraints
  • select one at random as the next guess
  • rinse-n-repeat until only one possibility remains

A simulation run was done for various initial guess choices. Each run fixed 1 or 2 initial guess words, and ran the simulator against all the GAME words. This was done 5 times for each initial guess and the results aggregated. I give 3 measures for comparison:

• GPW = average guesses per game

• total game score: 5 points for solved at guess #2, 4 points for guess #3, etc. No points for failure to solve in 6 guesses.

• weighted score: 16 points for solved at guess #2, 8 points for guess #3, 4 points for guess #4, etc.

I will report the results in the next post.

[Mathimagics]

I tested various 1-guess and 2-guess approaches. In the table below, GPW is average guesses to finish the game, S is the linear score, WS is the weighted score. Simulation ran 5 x 2315 = 11575 games for each initial guess choice.

Code: Select all

 # guesses =>     2     3     4     5     6    7+       GPW       S      WS
 --------------------------------------------------------------------------
 RAISE          647  1655  3503  2956  1684  1125     5.135   27960   45200
 CRATE          755  1801  3582  2957  1502   973     4.949   29141   48232
 TRACE          734  1776  3643  2948  1519   950     4.933   29118   47939
 TRAIN          665  1711  3529  3073  1624   968     5.000   28526   46214

 RAISE CLUMP         4351  3141  2474  1041   558     4.393   32841   53441
 SATIN HORDE         4723  3066  2224   983   569     4.336   33546   55559
 TRIPE COALS         5093  3129  2025   845   478     4.201   34654   58155


RAISE and CREATE both appear regularly in other people's recommended words list.

Ian Gowers, a mathematics professor at Cambridge, suggested choosing two words with a lot of commonly used letters where none are reproduced, for example “tripe” followed by “coals”.

And this in fact was the overall best performer for Marvin.

6-letter results

I naively assumed that a 6-letter game would be harder, after all, the DICT size is almost doubled. So I was somewhat taken aback by Marvin's report:

Code: Select all

 # guesses =>     2     3     4     5     6    7+       GPW       S      WS
 --------------------------------------------------------------------------
 RAISED         166  2292  3621  1983   773   780     4.801   25600   40215
 TRACED         216  2374  3727  1967   682   649     4.624   26373   41972
 DECANT         226  2327  3844  1889   688   646     4.621   26436   42074

 BANGED PURIST       3339  4154  1420   432   270     4.109   29090   46600
 COMING PASTER       2988  4233  1524   487   383     4.266   28186   44371
 CHINOS GARBLE       3081  4301  1544   426   268     4.146   28741   45366
 LASTED PORING       2940  4200  1562   511   402     4.296   27995   43955
 RINGED COSTAL       3202  4161  1388   474   395     4.236   28541   45510


I should point out that my target pool for 6-letter words has only 1924 words in it (and 23033 words in the DICT), so the scores look lower because I didn't scale them.

In any case, it's the GPW results that are interesting. Marvin finds the target with fewer guesses on average than for 5-letter Wordle.

This highlights a fundamental truth, I think - that the more letters there are in the game words, then the more information content there is to be had from the initial guesses.

My conclusion: 6-letters is in fact easier than 5, and if I were to build a game app for 6-letter play, I would seriously consider making 5 guesses the cutoff point!

[Mathimagics]

.
Some sample games (6 letters) from Marvin, with 2 fixed guesses. This numbers on the right are the number of possible words (NPW) after each guess.

Code: Select all

  M A N G L E     . Y G . . .     80
  P U R I S T     . G Y . . .      2
  K U N K A R     . G G . G Y      1
  R U N W A Y

  M A N G L E     . Y G . . .     80
  P U R I S T     . G Y . . .      2
  R U N W A Y     G G G G G G      1


Code: Select all

  M A N G L E     . G . . . .    532
  P U R I S T     . . . Y Y .     26
  Z A I D A S     . G Y . Y Y      3
  F A S C I A     . G G Y G G      1
  C A S S I A

  M A N G L E     . G . . . .    532
  P U R I S T     . . . Y Y .     26
  H A I C K S     . G Y Y . Y      1
  C A S S I A


[coloin]

Greetings Oh My !

I have my first word worked out .....
and then worked out before hand the next word options depending on vowels / consonants and how many letters ....
It depends if you want to try to do it in 2 [get lucky with first word] or play safe with an unrelated [to the first] word

scores
3-3
4-8
5-4
6-1

so my GPW average is 3.9 given that i have been improving as time goes....

[Mathimagics]

so my GPW average is 3.9 given that i have been improving as time goes....

With no failures assumed, that's 16 games, so GPW should be 4.2 [ (3*3 + 4*8 + 5*4 + 6*1) / 16 = 4.1875 ]
Marvin thinks that you divided by 17 instead ... tsk, tsk, shame on you !

I have my first word worked out ..... and then worked out before hand the next word options depending on vowels / consonants and how many letters ....

Is it like Quantian's method? For a fixed initial word (RAISE) he computed the next best word based on the YGx result and produced a table of next word choices.

Under optimal play, the hardest word to find is BOXER: you guess RAISE, then NOTED, then WHELM, then FARCE, then JOKER, and finally BOXER. Average word takes 3.56 guesses.

It is hard to know what "optimal play" refers to, as no details are given. I implemented his 2nd-guess table (based on first guess = RAISE), and for this option Marvin gets a GPW of 4.58, which is quite similar to the results for fixed 2 guesses like TRIPE + COALS, and is well short of Quantian's GPW = 3.56 !!

So, I am trying to work out what Quantian's "optimal play" method might be from the list of guesses he gives for BOXER.

[1to9only]

Interesting reading: https://medium.com/@tglaiel/the-mathematically-optimal-first-guess-in-wordle-cbcb03c19b0a
A quick summary:
- There are 2315 possible solutions, and 12972 words it accepts as guesses.
- ROATE is the optimal first guess, if the metric you’re measuring is “the size of the possible solution list after a guess”.

[Mathimagics]

Thanks 1to9only, an interesting read.

I see further down that he eventually decides that RAISE is best ...

I want to evolve Marvin (he's so depressed) into a "true bot", and to evaluate which approaches are actually more likely to win in general play - see Wordle Botfight - where a bot needs the ability to perform well for ANY target word in the list of 12972.

I see that the competition is stiff, with the leaders obtaining getting close to 4 GPW ... that's pretty impressive.

[coloin]

Thankyou will read it .... is IRATE up there too i wonder ?
And has anyone got the words .... asking for a friend !

[1to9only]

I've saved the Wordle web page, made some modifications (getting rid of some analytics stuff), and I play the game locally (offline) - there is no saved game stats after solving a wordle.

Another interesting article: https://bert.org/2021/11/24/the-best-starting-word-in-wordle/.
This one has a link to a 'beautified' version of the javascript code that drives Wordle - the list of solution words and acceptable words are in the script.
Direct link to beautified script: https://bert.org/assets/posts/wordle/main-beautified.js (right-click on link and Save linked content as...)

[Mathimagics]

So there you have it, the ideal first guess for Wordle is ROATE. It means:

“The cumulative net earnings after taxes available to common shareholders, adjusted for tax-affected amortization of intangibles, for the calendar quarters in
each calendar year in a specified period of time divided by average shareholder’s tangible common equity”.

A simpler definition, according to my Collins Scrabble Dictionary (2019):

ROATE (Shakespeare) to learn by repetition, also ROTE

[Mathimagics]

Oh dear, I was avoiding mention of those lists until it became clear that nearly everybody had them ...

... but its seems that the horse has long since bolted

[coloin]

Is this better play than deep blue

analysis and comments please .....

spoiler alert though as its todays 25/1

Hidden Text: Show

Code: Select all

IRATE  OYYOO
BONUS  OOOYY
SUGAR  GGGGG


[Mathimagics]

I have a WLD (Wordle Lie Detector) which sanalyses the # of possible words matching (NPW) after each guess, and then gives a carefully considered analysis of the results:

Code: Select all

  I R A T E     . Y Y . .     60
  B O N U S     . . . Y Y      1
  S U G A R     G G G G G

WLD:  Fail!


Code: Select all

   
  T R I P E     . Y . . .    123
  C O A L S     . . Y . Y      3

Possible words:
 1: SUGAR
 2: MARSH
 3: HARSH

WLD: Pass!


Note that WLD results are currently not admissible in court

[coloin]

I stand here on the steps of justice and fully respect the statutes of the Wangmeroo Legal Directive. I will reflect on the decision by by the jury and will take the required 24 hours to consider going down the appeals process. Should I chose the BONUS again I will be safe in the knowledge that TRIPE is not served for breakfast in Alcatraz

[Mathimagics]

Duly noted ...