Post by Kinwoods in Working out process

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Interesting. Today's (spoilers ahead) was a number, and something I found is that, while it doesn't give "is prime" for direct prime numbers away, it does when it's more than 2 multiples away:

395 (376 + 19) gives "x+395",
414 (376 + 19*2) gives "largest divisor is 2"
433 (376 + 19*3), however, gives "433 - x is prime". As does 319 (376 - 19*3).

Tangentially, I'm pretty confident in saying that the "largest divisor" clue is the 2nd highest priority (checking any, bar the nth powers, even number today (Oct 10) gives you that clue).

Are you sure that guessing only possible answers is a good idea? I guessed (499, 253) and got the same hint, which was amusing, but also quite unhelpful. Perhaps I should've tried an even number?

My only other discovery just now is that if I am 1 away from the solution, the game breaks.

That certainly is an... interesting bug.

luvrhino1 year ago (1 edit) (+1)

For 470 (376 + 94), I get "x is a multiple of (470-x)." It says that rather than "largest divisor is 94." Same is true for 423 and 564.

752 yields this: "x<752; (752-x) is a power of x,"

That is a new one for me. I had never tested double the solution before. Otherwise, that clue would be difficult to encounter, though you could have eventually starting with 512 & 243. Going the other direction, for 188, I get "x is a multiple of 188." Same for 94 and 47. I hadn't seen those either.

As for when I played for real, I did use (499, 212) because of a combination of eliminating 458 (as I'd get a "(499-x) is prime" clue) which made 212 the midpoint and because getting a "largest divisor clue" wouldn't be that bad at that point. That worked perfectly as it returned "x>212; x and 212's largest divisor is 4," which left 376 as the only possible solution.

If it were "largest divisor is 2," then I would have had 294 as the only solution more than 212 since 458 was already eliminated.

Going the other direction with x<212, the "largest divisor is 4" -> 048. "Largest divisor 2" -> 130. "(212-x) is prime" -> 171. "(212-x)'s largest prime divisor is 41" -> 007 or 089, though 089 might give "(x + 212) is prime."

I did not figure all that out live before I guessed. I just knew that getting a largest divisor clue would have been quite useful, which made 212 appealing beyond it being the midpoint among remaining candidates.

Kinwoods1 year ago (2 edits)

For 470 (376 + 94), I get "x is a multiple of (470-x)." It says that rather than "largest divisor is 94." Same is true for 423 and 564.

Ah, interesting. Guess I didn't check enough numbers, then.

752 yields this: "x<752; (752-x) is a power of x,"

I can see that hint being really useful, because that's basically an insta-win for anything above 249, which is 3/4 of the choices. But the likelihood of even getting that hint is quite rare.

Otherwise, that clue would be difficult to encounter, though you could have eventually starting with 512 & 243. Going the other direction, for 188, I get "x is a multiple of 188." Same for 94 and 47. I hadn't seen those either.

Hmm, after a bit of testing, I can't find any other hints. Think that's the lot of solutions then.

As for when I played for real, I did use (499, 212) because of a combination of eliminating 458 (as I'd get a "(499-x) is prime" clue) which made 212 the midpoint and because getting a "largest divisor clue" wouldn't be that bad at that point. That worked perfectly as it returned "x>212; x and 212's largest divisor is 4," which left 376 as the only possible solution.

Wish I were that lucky, cause I got the same clue for today (499, 233) and likewise for when I retried with (499, 200). Edit: I think I've been getting unlucky with your strat (I don't want to seem like I'm doubting your strategy, but genuinely (512, 243) would've given 3s for yesterday and today)

luvrhino1 year ago(+1)

Wish I were that lucky, cause I got the same clue for today (499, 233) and likewise for when I retried with (499, 200). Edit: I think I've been getting unlucky with your strat (I don't want to seem like I'm doubting your strategy, but genuinely (512, 243) would've given 3s for yesterday and today)

Today, I went (499, 246, 407). With 499, I had (499-x)'s largest prime divisor is 23, so I subtracted 11 * 23 = 253 to get 246, which is divisible by 2 and 3. Since I it returned the largest prime divisor is 23 clue again, I thought it was most likely to be 5 * 23 or 7 * 23 away. Otherwise, I'd get the "x and 246's largest divisor was 2" type clue. I was lucky enough to chose the latter.

However, yes, this is something that requires experimenting over time. You're correct that (512, 243) narrows today's down to a single solution, which makes it at least 0.5 guesses better than (499, 246) for today's number.

Except by accident, I have *not* played with having second guesses that aren't possible solutions. There may be certain clues where that's the optimal solution, but instinctively that feels wrong. However, that may also depend on what one's goal is...to minimize the average number of guesses or to minimize the number of losses. Those could yield slightly different strategies.

That does affect how I play Wordle, Dordle, and Quordle. Because I play against others, I used strategies that are optimized to beat my two friends at the slight increased risk of losing the game. Most notably, I stopped using ADIEU as a starter, because that made getting it in two extremely difficult and three less frequent than four.

For Dordle and Quordle, I do sometimes use words that aren't possible solutions to more efficiently eliminate letters. For example, I'd guess if the game returns -OUND, I might guess BUMPH to test four candidates for the first letter...even though that's British slang and would never be the solution. So, I'm open to something like your 243/729 second guess being best. I just feel it isn't for the reasons described elsewhere in the thread.

I will continue playing it both ways to compare.

Kinwoods1 year ago (2 edits)

Today (10/13) I got "x - 499 is prime". This, as I've said up-thread, I dislike, not because there are a lot of solutions (there's "only" 74), but because they're not easy to find (try determining if 708 - 499 is prime for example). I then try 729 'cause I've defaulted back to my old strategy, which is also just as (not) useful (I'm left with ~17 solutions), then do 601. 601's hint is "largest prime divisor" and helps me get the answer

But like, trying (512, 729) immediately gets it down to 6 solutions (and they're both the largest divisor hint!). It's not "better", because it's still extraordinarily unlikely to get a 3rd guess win, but you can see why I would prefer 512 in this specific case.

Though with all that said, I'm not giving up on the 499 start, despite its difficulty. From freeplay testing, it's better most of the time.

When you get the "is prime" hint, what do you do?

Today, I went (499, 246, 407). With 499, I had (499-x)'s largest prime divisor is 23, so I subtracted 11 * 23 = 253 to get 246, which is divisible by 2 and 3.

What I get from this is that I ought to guess even numbers after 499, or at the very least, find numbers that are of the form "499 ± a_semiprime_number"

So, I'm open to something like your 243/729 second guess being best. I just feel it isn't for the reasons described elsewhere in the thread.

That's fair enough; as a former (512, 243/729) player, I usually only got even numbers as remaining candidates, so I felt obligated to go for the prime hint (as an example, had I played today's like I used to, I'd probably have done (512, 729, 601, 588). 601 in particular because it's easy to calculate for stuff like "x + 601's largest divisor is 137"). So I agree it largely depends on perspective.

There may be certain clues where that's the optimal solution, but instinctively that feels wrong. However, that may also depend on what one's goal is...to minimize the average number of guesses or to minimize the number of losses. Those could yield slightly different strategies.
[...]
Because I play against others [...]

I like to minimize losses, then. My opening move for those games is (STORY, ADIEU), and so perhaps predictably, I average 4.6 guesses and have a 90% win rate (dunno if the winrate is better than avg or not tho).

luvrhino1 year ago

For 10/13, I was lucky again. Based on the previous discussion about possibly wanting to have factors for my second guess, I ended up using 732 as my next guess. "499 + 732" is a prime number, so that I felt like 732 was a legitimate candidate. Previously, I would have been more likely to use 690 or 692, because the distribution of prime numbers between 1 & 500 is more concentrated in lower numbers. If I wanted to divide them in half, the median number is likely around 181 or 191.

Also, in retrospect, picking a number that has a lot of factors seems like a bad idea, because I don't think I want "x and 732's largest divisor is {2, 3, 4, 6}," etc. Those are good after starting with "(x-499)'s largest prime divisor is 23" as it's already been narrowed down quite a bit. After starting with "(x-499) is prime," that clue would feel like trouble.

As it turned out, (499, 732) gave me "(732-x) is a square," which was great. Possible solutions were, at most, 732 minus {2², 6², 12², 14²}. It couldn't be minus an odd-number squared, because those numbers are odd, which means they'd fail the first clue. 8² & 10² also cause x - 499 to be composite. 4² wouldn't be a square, but a 4th power.

I went with 732 - 12² = 588, which was correct.

Again, I'm not sure what the best second guess is. If I want it to be a possible solution, it has to be even, which risks the dreaded "x and y's largest divisor is 2" clue. The prime numbers between 181 and 199 seem unlikely solutions because "499 + x" is not prime. If the solution were 680, for example, instead of "(x - 499) is a prime," I think it would give "x + 499's largest divisor is 131". as the clue. I haven't exhaustively searched all the options around that.

When I have more time, I might think about what to do here. Fortunately, that initial clue doesn't happen often.

I like to minimize losses, then. My opening move for those games is (STORY, ADIEU), and so perhaps predictably, I average 4.6 guesses and have a 90% win rate (dunno if the winrate is better than avg or not tho).

Average win rate for Wordle is probably a bit lower than 90% as some people don't really try that hard. I've been even more lucky there than I am in xdle as I've only lost once in 652 attempts. My first game was Jan 1, 2022. My average is under 4. However, I have an absurd memory for retaining words that have been used already. My memory isn't perfect, but it does allow me to navigate potentially fatal situations like getting SHA-E in all green tiles. I play with Hard Mode, so that means I could only test one letter per guess from that point forward. However, since I remember SHAKE and SHAME have been used, I might be able to salvage that.

I could not salvage -O-ER with Hard Mode and lost to FOYER. FOYER was obnoxious as the F & Y weren't shared with any other remaining solution of the form -O-ER. Same was true for BOXER, and JOKER, which were also candidates for me on my 6th guess. Those needed to be tested individually. I subsequently made changes to make that problem more rare. Using RAISE as my starter was one of those changes.

Kinwoods1 year ago

No comment for today's (10/14), except for the fact that I squandered an easy 3 by making several mistakes. Oh well.

Based on the previous discussion about possibly wanting to have factors for my second guess, I ended up using 732 as my next guess. "499 + 732" is a prime number, so that I felt like 732 was a legitimate candidate.

I would once again like to point out my unluckiness in choosing my numbers, for I am the cursed boy: while today I used (499, 729), when trying in a private browser with (499, 760), I got the same clue. Helpful-ish (I know it's odd, at least), but that then makes it a toss-up between the two remaining. Not so for (499, 729).

I play with Hard Mode, so that means I could only test one letter per guess from that point forward.

Oh yeah, my strategy would obviously not work on hard mode lol

I've been even more lucky there than I am in xdle as I've only lost once in 652 attempts

And all in hard mode? That's impressive.

However, I have an absurd memory for retaining words that have been used already.

I expect nothing less from someone who sees "x>499, 499 + x's greatest divisor is 53" and thinks "yes I calculate and keep track of all 9 candidates in my mind" :p

luvrhino1 year ago

Today, I did get it in three with (499, 760), which is at worst a coinflip, but may only have one remaining candidate.

Subtracting from 760, 1 * 29 and 8 * 29 are out because they're a prime number away from a clue. 2 *, 4 *, 5 *, and 6 * 29 are out because they'd yield a "Largest factor is <something>" clue, instead. I thought 7 * 29 = 557 would have yielded "(x+760)'s largest prime divisor is 439." Meanwhile, 673 + 760 = 1433, which is prime.

I still don't have a firm grasp on when the "(x + <guess>)'s largest prime divisor is <something>" clue appears.

For example, with today's 673 solution, if the guess is 29 off, it doesn't give the "(x-673) is a prime number." Instead it gives these two: "(x+644)'s largest prime divisor is 439" and "(x+702)'s largest prime divisor is 11." However, for 499, it gave us the "(x-499)'s largest prime divisor is 29" instead of "(x+499)'s largest prime divisor is 293."

I may be overlooking something obvious.

I expect nothing less from someone who sees "x>499, 499 + x's greatest divisor is 53" and thinks "yes I calculate and keep track of all 9 candidates in my mind"

My mental arithmetic ability is a much more of an outlier than my memory. This game enables me to exercise that ability. My brain is a bit weird in that it took me about as long to figure out 760 - 87 + 760 = 1433 as it did to figure out 1433 was prime. In cases like the "keep track of all 9 candidates in my mind," I am keeping track of them as 1 through 9 rather than 53 through 477...and I can usually eliminate several straight away.

Much of my mental math skill is based on minimizing how much of my very good, but not extraordinary, memory is required. For that reason, had I started today with 702 and had "(x+702)'s largest prime divisor is 11," I would have been very annoyed. My process would be:

702 mod 11 = 9
(2 * 702) mod 11 = 7, since 18 mod 11 = 7
This means the first candidate is 702 -7 = 695.
My guess will be 695 - 11n, for some value of n that cuts the remaining numbers in half...so I want n around 31.
(702 + 695) = 1397 = 127 * 11.
If (x + 702) were 90 * 11, that would be easy and fulfill the largest prime divisor being 11.
127 - 90 = 37
695 - 11 * 37 =
695 - 407 =
288
Ooh, that's nice because it has a lot of factors, since 288 = 17² - 1² = (17 +1)(17 - 1) = 18 * 16 = 2⁵ * 3² -> 18 factors
(702 - 288) nor (702 + 288) are perfect powers. Let's use that.

288 may not have been the optimal choice, but it feels like a valid solution. Moreover, it minimized the difficulty of the mental arithmetic.

For step 6 & 7, I could have chosen n = 31, which would have gave me 354 instead 288. That's closer to the midpoint between all candidates, 2 through 695. My brain decided to simplify the step where I made (x + 702) = 90 * 11 instead of 96 * 11, even though I didn't need to calculate that. 354 probably is the slightly better choice.

Anyway, my point in sharing all that is how, again, I can do all this while minimizing the memory requirements and how difficult the arithmetic is.

luvrhino1 year ago

BTW, you probably know this, but your math ability exceeds that vast majority of the world. That's probably why you were drawn to this game in the first place. However, your mathematical instincts and ability to understand what the heck I'm talking about is very impressive.

It has not gone unnoticed or unappreciated.

Kinwoods1 year ago (3 edits)

However, your mathematical instincts and ability to understand what the heck I'm talking about is very impressive.

Thank you! Nice to know that going to a STEM school pays off for something. :)

if the guess is 29 off, it doesn't give the "(x-673) is a prime number."

Ye, I've noticed it doesn't give it when it's a prime away from the target number (more on that at the end).

The process

Of course! Using modular arithmetic to not have to calculate large numbers is genius! Especially since you're not actually needing to calculate all the candidates, just the center ones. For step 4, are you approximating the value (i.e. 695/11 ≈ 695/10 ≈ 70 => I need n ≈ 35 )?

Step 11 and 12

How high do you know your squares? While I can see how 702 ± 288 are obviously not squares (the sum ends in 0 => check if 30*30 or 40*40 is equal to 990 meanwhile the difference is 414, so not 20*20 = 400 and not 21*21 either else it'd end in a 1), I'm still interested in knowing. Me personally, I know up to 16, 20/30/40/etc, and 25.

I may be overlooking something obvious.

I think "x+guess" appears when "x-guess" doesn't (and when other hints don't take priority). I think. This is pure speculation.

However, I can tell you when "x-guess" doesn't appear: if x - guess = largest_prime (Edit 2: and x + guess = not_a_prime). For example: today (10/16) I started with 499, got "largest prime divisor is 43". Using pen and paper, I find that 800 is a possible candidate, is close to the center, and frankly easy to do calculations with.

Now, 800 also gives me "largest prime divisor is 43", but this time, instead of complaining, I actually take the time to think about it. I know that x > 800, so the remaining possibilities are (843, 886, 929, 972). As previously mentioned, it's not 843 ~~or 886~~, else it wouldn't have given me the 43 hint. It's not 972 (Edit: or 886) either, else I'd've gotten "x and 800's largest divisor is 4". That leaves only 929 (which is indeed today's answer). Pretty neat, right?

Edit: Uh, whoops, largest_prime*2 is nonsense.

To further my point:

929 - 700 = 229. That's prime, so we get x+700.

~~929 - 843 = 86 = 43 * 2. That's prime*2, so we get x+843~~. Oh wait no we don't

929 - 886 = 43. That's prime, so we get x+886.

929 - 600 = 329 = 47*7. So we get x-600.

929 - 800 = 129 = 43*3. We get x-800.

Edit 2: here's something interesting: if both x-guess and x+guess are prime numbers (try 918 for example), it gives x-guess is prime.

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