Letters Max Progression (Featuring graphs)

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JackHurst
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Letters Max Progression (Featuring graphs)

Post by JackHurst » Sat Oct 17, 2020 2:11 pm

Here I share some analysis (with help from Rob Foster) of how much effort one has to put in to progress from complete n00b to letters
God.

Hypothetical Scenario

Suppose we have a person called Alice. Her lexicon starts as completely empty, but she can learn words consistently with perfect recall. That is to say:
  • It takes a fixed amount to effort to learn a word and then it stays in her lexicon forever
  • When she plays countdown you can assume she will spot all of the words in her lexicon without fail and will never declare a word not in her lexicon
  • Therefore she spots the max in a round if and only if the maxes list contains a word she has learned
(This makes her a bit like Apterous Sponge)

Suppose Alice has access to a copy of the Countdown lexicon sorted in a way optimized for efficient learning:
  • For each anagrammatic set, only one word is included
  • The top word is the word most likely to appear as a max in all countdown round
  • The second word is the word most likely to appear as a max in all rounds which don't also have the first word as one of the maxes
  • The nth word is the most frequently occurring max in all countdown rounds which don't contain any of the first n-1 words as a max
Some Stats

The list produced by the above method starts like this

Code: Select all

Rank,Letters,Word
1,AEGIORT,GOATIER
2,AEINOST,ATONIES
3,ADEIORT,RATIOED
4,AEENOSU,AENEOUS
5,AEIORSTU,OUTRAISE
6,AEEGOST,GOATEES
7,ADEIOST,IODATES
8,ADENOSU,ADENOUS
9,AEEILNO,AEOLINE
10,ADEGIOT,GODETIA
The full Countdown lexicon of around 160K (need to get citation for that) is reduced to around 89K words. Most of that saving is due to wiping out anagrammatic sets, and then you get extra savings from words that are impossible to come up because additional vowels would always make a longer or equally long but more likely word to come up (e.g. CANTER) and you also have words like PIAZZA which can never come up because there is only 1 Z in the letter pack.

So if Alice goes about learning the lexicon in this efficient manner, how will her letters max go up as she progresses through the lexicon? here's the graph up to 90%:

Image

Interesting is that the progression looks almost logarithmic

For Alice, to gain each successive letters % point it becomes increasingly harder. She gets diminishing returns.

What about the real world

In reality humans don't function like the above and we can all think of many many reasons why the above model is completely unrealistic for how a real human would progress at countdown. Hopefully you can all see one thing though which is the diminishing returns. Learning 100 new words when you are a 50% letters player gives you more bang for buck than learning 100 new words when you are a 70% letters player.

Let's look at the superstats letters graphs for some players who have got to God level (>80% maxes).

Player 1

Probably my favourite graph of them all. A player who started just under 10 years ago as a complete novice, has continued to play for most of those ten years showing the logarithmic growth. Also fun to see that this player wasn't deterred by the dictionary update around 1700 days ago.
Image


Player 2

A player who joined 11 years ago as a rookie and has pretty much stuck around making consistent progress ever since with growth that also looks logarithmic. A notable period of absence around 1700 days ago coinciding with the big dictionary update. My bet would be that this player got fed up around that time and needed a bit of time off. Overall a much quicker recovery from the dictionary update than player 1. Perhaps this player studied high probability word lists of the new words, whereas player 1 was happy yo just plod along on apterous and pick things up as they came along?
Image

Player 3
A player who joined 4 years ago, already as an intermediate. Steady progress to God level in about 2 years and then diminishing returns since. I wonder was this player just naturally very good when they joined or did they have a different account first? I find either equally plausible. The think I like about this graph is that this is the only example I could find of a God level player who joined after the dictionary update.

Image

Player 4

A steady progression into a very solid player for player 4. I really like the consistency in this one. I'd guess they've plugged away at apterous for years and gotten pretty darn good from it. It looks like their growth has been minimal in the past three years though. Maybe they need to start drilling word lists if they want to get to the dizzying height of God level.

Image

Player 5
Another standard logarithmic progression and 1700 day dip for a long standing player who reached an impressive standard. I'm not quite sure this player is God tier either, but if they came back and played for another couple of years as much as they used to I have no doubt they could get there.

Image


Conclusion
I had quite a bit of fun doing this and hope it triggers some interesting discussion. There's loads of data out there, and probably quite a bit of stuff that disagrees with the hypothetical model a lot less. People are imperfect and erratic and that will lead to many different shapes of graphs. Points I would say:
  • Initial growth is pretty rapid. Most players seem to get 20% - 30% added on in their first 18 months
  • Growth slows down as time goes on
  • The dictionary update in early 2016 made the game a lot harder to max
  • Getting to God level appears to take at least 2-3 years of solid progress and hard work
  • Once at God level growth is almost non existent (do people just give up because the investment required to get any better is more than they are prepared to give?)
  • It still appears to take years more hard work to get from elite level to god level

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Charlie Reams
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Re: Letters Max Progression (Featuring graphs)

Post by Charlie Reams » Sat Oct 17, 2020 2:55 pm

Good stuff. Interesting that maxes are such a good proxy for overall performance.

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Rhys Benjamin
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Re: Letters Max Progression (Featuring graphs)

Post by Rhys Benjamin » Sat Oct 17, 2020 4:44 pm

(looks at my letters max graph... never mind)

But yeah this is bloody brilliant.
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Fiona T
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Re: Letters Max Progression (Featuring graphs)

Post by Fiona T » Sat Oct 17, 2020 4:54 pm

Blimey, there's 3 words on that top 10 list I've never played. Clearly need some intelligent focus to my sporadic learning activities.
8-) <-2m-> 8-)

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Re: Letters Max Progression (Featuring graphs)

Post by Sam Cappleman-Lynes » Sat Oct 17, 2020 4:56 pm

I think the graph is probably closer to something like 1 - k^n for some k slightly smaller than 1, rather than something like log(n).

At least, this is true under the (completely false) simplification where each word appears as a max with fixed probability independently of all other words, and with the assumption that, apart from the head and the tail, most words' probabilities fall in a fairly narrow range.

JackHurst
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Re: Letters Max Progression (Featuring graphs)

Post by JackHurst » Sat Oct 17, 2020 5:21 pm

Sam Cappleman-Lynes wrote:
Sat Oct 17, 2020 4:56 pm
I think the graph is probably closer to something like 1 - k^n for some k slightly smaller than 1, rather than something like log(n).

At least, this is true under the (completely false) simplification where each word appears as a max with fixed probability independently of all other words, and with the assumption that, apart from the head and the tail, most words' probabilities fall in a fairly narrow range.
Yeah this makes perfect sense. Nicely explained :)


By the way, bonus points for those who can correctly identify the graphs above.

Sam Cappleman-Lynes
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Re: Letters Max Progression (Featuring graphs)

Post by Sam Cappleman-Lynes » Sat Oct 17, 2020 5:34 pm

JackHurst wrote:
Sat Oct 17, 2020 5:21 pm
By the way, bonus points for those who can correctly identify the graphs above.
Players 1, 2 and 3 are Jack Worsley, Rob Foster and Elliott Mellor. Working on the others!

Edit: Player 4 is Jonathan Wynn.
Edit 2: Player 5 is Zarte SIempre.

Thomas Cappleman
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Re: Letters Max Progression (Featuring graphs)

Post by Thomas Cappleman » Sat Oct 17, 2020 6:18 pm

Fiona T wrote:
Sat Oct 17, 2020 4:54 pm
Blimey, there's 3 words on that top 10 list I've never played. Clearly need some intelligent focus to my sporadic learning activities.
The list does heavily favour words with lots of vowels, which is then only useful if you or your opponent pick 5 vowels a lot. I think because a given selection with 5 vowels is more likely than similar 4 vowel selections, as there's only 5 vowels to choose from rather than 21 consonants, so maxes for those selections are more useful.

Gavin Chipper
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Re: Letters Max Progression (Featuring graphs)

Post by Gavin Chipper » Sat Oct 17, 2020 6:19 pm

Nice work. I don't have superstats, but I'd be on this God list otherwise.

JackHurst
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Re: Letters Max Progression (Featuring graphs)

Post by JackHurst » Sat Oct 17, 2020 6:27 pm

I only simulated 10000 rounds and yeah 3 4 or 5 vowels were with equal probability.

Gavin Chipper
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Re: Letters Max Progression (Featuring graphs)

Post by Gavin Chipper » Sun Oct 18, 2020 8:01 am

And the moral of this story... :o

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Ian Volante
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Re: Letters Max Progression (Featuring graphs)

Post by Ian Volante » Sun Oct 18, 2020 12:40 pm

Interesting to see my stats compared to those above: the dictionary update caused me trouble, but recent incentive has helped me approach sixty percent for the first time at least.

Image
meles meles meles meles meles meles meles meles meles meles meles meles meles meles meles meles

Fiona T
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Re: Letters Max Progression (Featuring graphs)

Post by Fiona T » Sun Oct 18, 2020 12:52 pm

Ian Volante wrote:
Sun Oct 18, 2020 12:40 pm
Interesting to see my stats compared to those above: the dictionary update caused me trouble, but recent incentive has helped me approach sixty percent for the first time at least.
Nice - bodes well!

Mine's weird - brief peak to 60% rapidly returning to normal! I've a feeling that might have coincided with my self-imposed 'dry July' :D

Image

Perhaps needs some work to get that spike of improvement seen by others
8-) <-2m-> 8-)

Dan Byrom
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Re: Letters Max Progression (Featuring graphs)

Post by Dan Byrom » Sun Oct 18, 2020 1:03 pm

I have a very similar spike Fiona!

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