I don't think the different usage is related to an individual group rather context, particularly when referring to numbers.
E.g. 3 is a 1 digit number because that's the number of digits needed to uniquely identify 3. There are infinitely more ways to identify 3 the number with padded 0s but those aren't useful unless you're talking about combinations/sequences of groups of digits (like random numbers or PIN codes) instead of actual number values.
While natural in terms of just inputting numbers, it's also not bad to use as first guess because primes are denser (in an "average absolute distance" sense) the lower the magnitude you're around. Without having checked, I would expect there to be more primes of the form 123** than 987**, so you're more likely to gain correct digits with this guess.
Edit: I checked, it's 9 vs 8 primes. But nearby prefixes also have more and there's a lot of variance. Still, there are overall more primes among the same range of smaller numbers (1000-5499 has ~4400 primes, 5500-99999 has ~4000).
Note that, despite this, you probably shouldn't include 0 in the starting guess (e.g. 10247) because 0 will be in the solution less frequently (can't be the leading digit), meaning you gain less info on average.
I read the "help" thing, which suggested 71429 as the first guess, so I started there. (9 was right, and the other digits were wrong. That meant the second guess got the right digits, and the third guess got the right prime.)
log(100000) is 5, so approximately 1 in 5 numbers below 100k is prime. Obviously those in the range 10000-99999 are less dense, but primes are still surprisingly common.
That being said, the pool you're really working from is the numbers with last digit 1, 3, 7, or 9. One out of every 4.6 such numbers under 10^5 is prime. So just guessing until you find a prime is practical.
Ah you're correct. Annoying how Google interprets log(x) as base-10 (but more shame on me for not realising that e^5 is obviously not 100000). In my experience everyone uses log and ln interchangeable outside of lessons at school.
> On calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when they write "log".
I intuited a probably-inaccurate trick with exponentially raising a small prime (e.g 7*7*7etc), doubling the multiple (*2), and subtracting 1 (to get an odd number).
That got me two substantially-different primes and narrowed my numbers a lot, which then became guesswork.
The number you come up with in that way is at least guaranteed to not be divisible by 2 or 7. Obviously that doesn't mean it's prime but it definitely helps your odds.
Also usefully, (2(a+1)^n-1) % a = (2a(a+1)^(n-1) + 2(a+1)^(n-1) - 1) % a = 1. So if you choose one of the 3k+1 primes, it's also guaranteed not divisible by 3.
apparently 9.3% of all '5 digit' numbers are prime, so random guessing isn't a bad strategy - you'll find one in 10.75 totally randomly chosen numbers to be prime.
Being a little smarter, prime number can only end in a 1, 3, 7 or 9 - (ending in 0, 2, 4, 6, 8 would be even, ending in 5 would be odd), so in fact it's more like 25% of 'likely' guesses.
IIRC, randomly guessing is _always_ a good strategy for "give me a prime in the range [a, b]" , in other words that's what's used for algorithms that need primes anyway. Either guess and check or guess and increase-by-2-until-prime.
Does work _much_ better in this range than at crypto sizes though.
I thought the same, but it rejects guesses that are not valid prime numbers. So you'll find some with trial and error. With that it took me only three guesses.
You can guess a prime relatively easily given a few attempts, as besides 2 and 5, all other primes will have a 1, 3, 7, or 9 as its least significant digit. So, for starters, you'd never guess a number with any other least significant digit.
All one- and two-digit numbers that look prime are prime, except 91.
This assumes that you can easily identify
- multiples of 2 and 5 (by their last digit)
- multiples of 3 (the sum of their digits is divisible by 3)
- multiples of 11 (for two-digit ones, they have both digits the same: 11, 22, 33, ..., 99)
- squares
So the first number that looks prime but isn't is the product of the two smallest primes that aren't "easy", which is 7*13 = 91.
Then recall that "less" includes a regex-based search on the / key.
This doesn't give you direction as to the best to try, of course, but it's hard to beat it in terms of bang for the buck if you're just going to try a few.
Yeah I would really rather this just let me guess non-primes even though that would make it more challenging. Trying to find a prime by guessing randomly over and over is pretty frustrating.
The intent, I think, is to mimic how Wordle works. Wordle has one puzzle per day, progress and statistics are stored in a cookie on your local machine (and only there). You can clear the cookie or use another browser or privacy mode to play multiple times per day, but you'll always get the same word.
It's a clone of Wordle. There is a single correct answer per day, and everybody gets the same puzzle. There will be a new answer tomorrow.
So, since the puzzle will be the same, there isn't a huge amount of reason to let you reset.
The page could have made that clearer. (Wordle says it when you open it the first time, and also under the help menu.) I double-checked this by looking at the sourcecode.
I thought coming up with primes would be very challenging, but then my first three guesses were all prime (all I did was pick numbers that ended with odd numbers that weren't five). I got it on the fourth after a few attempts at entering the number - the prime constraint here makes it easier.
There are 8,363 five digit primes. If you limit your guesses to numbers ending in 1, 3, 7, and 9, there is a 23% chance of randomly picking a prime.
Amusing to see people starting with odd digits so much. You're going to have to use an odd digit in every guess, so you can more efficiently explore the space by going after all the even digits first with 24683.
Looks very nice. However it took me a while to figure out why I kept getting error. It wasn't obvious to me that you are required to enter prime numbers only. Maybe consider dropping this constraint?
Also, might want to disable text selection via css because I kept selecting the numbers on the button on my phone.
I couldn't come up with any primes, nor I'm any good with numbers, so I started with the prime from the game example:
71429
Than I already had one digit guessed and simply clicked the unused numbers from the virtual numpad below and got to having all digits with just two being in the wrong place:
I got a popup 'Not a 5 digit prime' on my first try and it wouldn't let me try any further. Not sure if I'm dense or if the UX is just terrible
I had tried 18141, by the way, which I modestly think was a pretty good guess in hindsight given its only two factors are 3 and 6047. 19141 would have been a prime, but perhaps not the one I was supposed to find? I don't play Wordle so I'm a little confused
My side-project this week has been searching for an optimal Wordle strategy. I figured it would make a great blog post but I haven't got that far yet.
Step 1 is to find an optimal starting word. My most insightful finding so far has came from trying to define a cost function for comparing potential starting words. It turns out that "% of potential guesses [not] eliminated" is an excellent loss function.
Importantly, the full set of all Acceptable Guesses is knowable. For any given (guess, answer) pair, the game provides feedback about each character (or digit). There are basically three pieces of potential feedback: (N)ot Used, (U)sed Elsewhere, (E)xact Match. For example, a single guess might produce the feedback "NUNNE". Each of these will eliminate some subset of Acceptable Guesses which means you can attribute a fixed value between 0.0 and 1.0 to any piece of feedback, and multiply those together to get the loss score of a given guess. Average that across all Potential Answers to get a cost score.
Not sure what my goal of this post is. Guess I just wanted to share that these games are as much fun to analyze as they are to play (if not more).
Since you've been looking into it, how are words with multiple of the same letters treated on Wordle. If there are two (or three) 'E' in the solution or the guess?
I've spent some time analyzing Wordle (my starting words are SOARE and then BUILT, unless I've already got strong clues from SOARE).
If the answer has one E and you guess a word with two Es, only one of the Es will be marked correct (green/yellow). If one of the Es is in the correct place, it will be green and the other one grey. Otherwise the first E will be yellow and the second grey. So if the answer has fewer Es than your guess the game tells you.
If the answer has more Es than your guess there is no indication of this.
I've given some light thought to this: There are several which are considered "good starting words", that mainly contain lots of vowels or most common used letters (such as "audio", Raise, etc). Then, as a second word, you could use another word that complements those letters (audio/rents, raise/mould ).
But there is a better strategy (similar to what is used in those interview puzzles of "From N weights, find 1 that is different in X measurements of a scale): The fact that you input some vowels and see that they are not in there, tells you that the remaining ones MUST be in the word (treating Y as a vowel as well). So it might be possible to come up with a strategy that uses that negative information as well to minimize the search space.
I use WEIRD, my completely untested and I researched intuition being that knowing common consonants are in a word isn’t helpful unless you get the right place, but knowing the vowels is. At least that was my logic when I first played and my brain produced WEIRD.
I’m sure it’s far from optimal, but I continue to use it because I obviously understand how I came to choose it, and it “works” and survivor bias is powerful :)
I recall seeing HN posts from people working out “optimal” starting words and guess patterns, but I just like mine :)
I made a lil clone of wordle and found that the full list of solutions, past and future, is right there in the minified js. Could be useful for your analysis.
Maybe I shouldn't be sharing this, it's a bit sad to break the illusion that the creator is picking a new word for us every day.
Today 538 gave the optimal starting word and all the best second words, with a 60% chance of solving it in three words. Completely ruined the game, in my opinion.
Interestingly, I made a coding challenge for this exact problem a few years ago. You can find it on the Code Golf StackExchange website: https://codegolf.stackexchange.com/questions/26858/guess-the....
The challenge is named after the Lingo TV show which used the concept repopularized by Wordle.
Lingo itself took inspiration from a 1955 paper game named Jotto.
Regarding strategies used in the challenge, the best performing solution was an adaptation of a Mastermind solver.
This website is currently hosting a competition until the end of the month for Wordle strategies if you want to see how it compares: https://botfights.io/game/wordle
I went to the source code of wordle & copied all 12000 words to an excel sheet Column A, starting from row 4.
3rd row for next 18 columns has drop down filter.
You guess any word with at least 2 vowels, if 3 better, no repeat letters. You get some grey, some yellow, some maybe green.
2nd column to 6th column on excel sheet is the Grey Letters, Not Found in Answer. In this, 2nd row, if u find any grey you type it here. The formula from row 4 downwards in this group checks if its top input cell is empty, if yes, true, if not empty, then it check if that letter exists in its row cell of column A. If exists, false (grey means no letter), otherwise true.
Column 7th to 11 are yellow. Same thing, row 2 gets input, 4th row onwards checks if this exists in Column A cell, true, otherwise false. If input cell empty, then True.
Column 12 to 16 gets green letter input. Here input goes by position, if 2nd letter is green, you type it in 2nd column of this group, which is 13th column.
Formula in 4th and next checks if input empty, true, otherwise if input exist in exact position, true, otherwise false.
17th column is empty & narrow, to create a gap.
18th column returns 0 if any false found in while row. Any false means this word is not the answer. Otherwise returns 1.
A separate one cell counts all those 1, tells me how many potential answers are. After every try/word, one can filter this last column on 1.
138 comments
[ 2.9 ms ] story [ 251 ms ] thread65537
12497
80747
23459
64217
14771
Apparently I have stumbled over an idiom of the prime-enthusiast community.
E.g. 3 is a 1 digit number because that's the number of digits needed to uniquely identify 3. There are infinitely more ways to identify 3 the number with padded 0s but those aren't useful unless you're talking about combinations/sequences of groups of digits (like random numbers or PIN codes) instead of actual number values.
It’s not a significant digit[0], though, when it’s leading.
[0] https://en.wikipedia.org/wiki/Significant_figures
But I still love the concept!
⬛⬛⬛⬛ ⬛⬛⬛ ⬛
Partially interpreted:
⬛⬛🟨⬛⬛
🟨⬛🟨⬛⬛
🟨🟨🟨⬛🟩
🟩🟩🟩🟩🟩
Edit, I guess HN doesn't like emojis lol
Edit: I checked, it's 9 vs 8 primes. But nearby prefixes also have more and there's a lot of variance. Still, there are overall more primes among the same range of smaller numbers (1000-5499 has ~4400 primes, 5500-99999 has ~4000).
Note that, despite this, you probably shouldn't include 0 in the starting guess (e.g. 10247) because 0 will be in the solution less frequently (can't be the leading digit), meaning you gain less info on average.
That being said, the pool you're really working from is the numbers with last digit 1, 3, 7, or 9. One out of every 4.6 such numbers under 10^5 is prime. So just guessing until you find a prime is practical.
> On calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when they write "log".
That got me two substantially-different primes and narrowed my numbers a lot, which then became guesswork.
My guesses in the end:
by the way am I supposed to take some meaning from the italicized '7' or was that just a typo?
Being a little smarter, prime number can only end in a 1, 3, 7 or 9 - (ending in 0, 2, 4, 6, 8 would be even, ending in 5 would be odd), so in fact it's more like 25% of 'likely' guesses.
Does work _much_ better in this range than at crypto sizes though.
This assumes that you can easily identify - multiples of 2 and 5 (by their last digit) - multiples of 3 (the sum of their digits is divisible by 3) - multiples of 11 (for two-digit ones, they have both digits the same: 11, 22, 33, ..., 99) - squares
So the first number that looks prime but isn't is the product of the two smallest primes that aren't "easy", which is 7*13 = 91.
Unlike Wordle, I assume it's unreasonable to expect a player to know primes so I don't think I'd call this cheating.
This doesn't give you direction as to the best to try, of course, but it's hard to beat it in terms of bang for the buck if you're just going to try a few.
The intent, I think, is to mimic how Wordle works. Wordle has one puzzle per day, progress and statistics are stored in a cookie on your local machine (and only there). You can clear the cookie or use another browser or privacy mode to play multiple times per day, but you'll always get the same word.
So, since the puzzle will be the same, there isn't a huge amount of reason to let you reset.
The page could have made that clearer. (Wordle says it when you open it the first time, and also under the help menu.) I double-checked this by looking at the sourcecode.
Good point! Somehow, being numbers, I thought it would just make up another.
Now come up with a hash that starts with many zeroes.
There are 8,363 five digit primes. If you limit your guesses to numbers ending in 1, 3, 7, and 9, there is a 23% chance of randomly picking a prime.
Also, might want to disable text selection via css because I kept selecting the numbers on the button on my phone.
Just say so!
71429
Than I already had one digit guessed and simply clicked the unused numbers from the virtual numpad below and got to having all digits with just two being in the wrong place:
35869
Now this was easy to figure out.
Got the following score: 20 3/6
⬜⬜⬜30029 ⬜⬜⬜⬜12347 ⬜⬜⬜⬜33331 ⬜⬜⬜⬜13337 86539 65839
I had tried 18141, by the way, which I modestly think was a pretty good guess in hindsight given its only two factors are 3 and 6047. 19141 would have been a prime, but perhaps not the one I was supposed to find? I don't play Wordle so I'm a little confused
1+8+1+4+1 = 15 15/3 = 5
I think it should clear the whole guess after an invalid guess.
1 2 [3] 4 7
[5][6](8)[9][3]
(6)(5)(8)(3)(9)
Fun game! Might take a while to explain to people though!
Step 1 is to find an optimal starting word. My most insightful finding so far has came from trying to define a cost function for comparing potential starting words. It turns out that "% of potential guesses [not] eliminated" is an excellent loss function.
Importantly, the full set of all Acceptable Guesses is knowable. For any given (guess, answer) pair, the game provides feedback about each character (or digit). There are basically three pieces of potential feedback: (N)ot Used, (U)sed Elsewhere, (E)xact Match. For example, a single guess might produce the feedback "NUNNE". Each of these will eliminate some subset of Acceptable Guesses which means you can attribute a fixed value between 0.0 and 1.0 to any piece of feedback, and multiply those together to get the loss score of a given guess. Average that across all Potential Answers to get a cost score.
Not sure what my goal of this post is. Guess I just wanted to share that these games are as much fun to analyze as they are to play (if not more).
If the answer has one E and you guess a word with two Es, only one of the Es will be marked correct (green/yellow). If one of the Es is in the correct place, it will be green and the other one grey. Otherwise the first E will be yellow and the second grey. So if the answer has fewer Es than your guess the game tells you.
If the answer has more Es than your guess there is no indication of this.
But there is a better strategy (similar to what is used in those interview puzzles of "From N weights, find 1 that is different in X measurements of a scale): The fact that you input some vowels and see that they are not in there, tells you that the remaining ones MUST be in the word (treating Y as a vowel as well). So it might be possible to come up with a strategy that uses that negative information as well to minimize the search space.
I’m sure it’s far from optimal, but I continue to use it because I obviously understand how I came to choose it, and it “works” and survivor bias is powerful :)
I recall seeing HN posts from people working out “optimal” starting words and guess patterns, but I just like mine :)
Maybe I shouldn't be sharing this, it's a bit sad to break the illusion that the creator is picking a new word for us every day.
https://aditya-sengupta.github.io/coding/2022/01/13/wordle.h...
3rd row for next 18 columns has drop down filter.
You guess any word with at least 2 vowels, if 3 better, no repeat letters. You get some grey, some yellow, some maybe green.
2nd column to 6th column on excel sheet is the Grey Letters, Not Found in Answer. In this, 2nd row, if u find any grey you type it here. The formula from row 4 downwards in this group checks if its top input cell is empty, if yes, true, if not empty, then it check if that letter exists in its row cell of column A. If exists, false (grey means no letter), otherwise true.
Column 7th to 11 are yellow. Same thing, row 2 gets input, 4th row onwards checks if this exists in Column A cell, true, otherwise false. If input cell empty, then True.
Column 12 to 16 gets green letter input. Here input goes by position, if 2nd letter is green, you type it in 2nd column of this group, which is 13th column.
Formula in 4th and next checks if input empty, true, otherwise if input exist in exact position, true, otherwise false.
17th column is empty & narrow, to create a gap.
18th column returns 0 if any false found in while row. Any false means this word is not the answer. Otherwise returns 1.
A separate one cell counts all those 1, tells me how many potential answers are. After every try/word, one can filter this last column on 1.