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
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
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
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%:
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).
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.
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?
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.
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.
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.
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