itch.io

(spoiler)

I got 33 nines in 82 cycles:

I wanted 8 or 16 for my third guess to add one or two more factors of 2 beyond just being divisible by 4.

Makes sense.

You made this more complicated than necessary given that you involved Wolfram Alpha.

I was explaining why I thought the way I did. I involved WA because I wanted to be sure that I didn't miss any candidates (and as you can see, I did). This is not how I was thinking when I was doing the xdle, and also I like to only use pen and paper when solving. When I played, it was more to the effect of "oh x-744 is square that means it has to be 744 + 49".

In hindsight, I should've just gone for something like an antiprime, since they would've been much more useful here.

I wanted it to be divisible by 8, if not 16.

Wouldn't you want to search for numbers divisible by 4? I must be missing something here.

[ETA: I just tried 475 when 192 was the solution (475 + 192 = 667) and did get "(x+475)'s largest prime divisor is 29" as I suspected.  My logic for choosing 192 over 168 was correct.]

Make sense, given that 475-192 = 283, which is prime.

I ended up getting lucky based on somewhat bad thinking.

Hey, same here! Today I tried (499, 744). 499 gives "largest prime divisor is 7", and 744 gives "x-744 is a square". Since we know that (x-499) mod 7 = 0, and also (744-499) mod 7 = 0, then that necessarily means that x = 7n + 499, where n > 35 is some natural number. Or, put another way, x = 7m + 744, m > 0 a natural. But we know that x - 744 = (7m + 744) - 744 = 7m is a square. Therefore, m = 7, and that's the only possible value it can take.

Except I get this feeling. This feeling that I'm Wrong. And indeed, WolframAlpha tells me that I'm Wrong, because the answers to "7m is a square" is m = 7(y*2). If y=1, then we do get m=7, but if y=2, then m=28... and uh oh, x = 7*28+744 = 940 < 999...

Sadly, I noticed that there were no posts about it after it crashed, so I went with 498 instead of 500 on the theory that people were slightly more likely to guess 501 over 497 and encounter the bug...or that someone would get it in 1 and post about it. 

I'd reckon not many. Xdle is at a very specific cross of (Wordle players) and (Math pros) is not large. Not to mention the complete inability to look this game up using itch.io's search bar ('cause Xdle is stylized with a cursive x), or the fact that Xdle is harder to remember (X is used as a variable, but if you don't make that connection you're out of luck).

That's basically a fancy way of saying I needed numbers where (x - 499) was divisible by 7 and where x was divisible by 2 and not 4.

Ah, okay, got it.

Unrelated, but curious to know what you did today (10/24). Today, though I did not use this strategy, is a particularly good example of how (512, 243) is occasionally useful. (FWIW, my guesses were (499,243,151,192))

Can you explain this "Wrong. 729 - x is prime, which means that 729 + x is also prime, per my last comment"?

Sure. Per my experimentation, xdle prefers giving the "guess - x's largest prime divisor" hint as opposed to "guess + x's largest prime divisor", which it prefers to "guess - x is prime". You can think of it like how xdle prefers "largest divisor" and "is an nth power" hints to "prime divisor" hints. I haven't seen evidence to the contrary, though I haven't exhaustively checked every number.

Furthermore, xdle won't give the "guess - x's largest prime divisor" hint if guess - x (or x - guess) is itself a prime. Instead, it'll give "guess + x". If guess + x it also a prime, it'll instead give you "guess - x is a prime". I can give some examples:

• 123: 123 + 500 = 629 = 7*89, but because 500 - 123 = 377 = 13*29, you'll get "x - 123's largest divisor is 13".
• 79: 79 + 500 = 579, which is prime, so instead you'll get "x+579's largest prime divisor is 83".
• 99: 500 ± 99 is prime, so you get "x - 99 is prime".

I'm not sure that "729 + 646 = 1375" would prompt a "x + 729's largest divisor is 11" clue.  Those clues seem to usually have the divisor be fairly large...though I don't know what the limits are for fairly large.

They're large because due to how the hints are ordered, you're more likely to get "guess-x" and "guess and x's largest divisor" over "guess+x",  but low divisor hints do exist. For example, try 529 or 67 for today (10/23). You'll get "x+guess's largest prime divisor is 7".

Which means it is also 499 +n * 7 where n mod 4 = 1

Could you explain how you came to this conclusion? I understood the rest, just having a hard time figuring out how you used modular arithmetic here.

Today's (10/23) xdle crashes with 499. (fake confusion) Hmm, I wonder what this could mean... (hey, it's my first 1-guess win!)

Today (10/21) is probably my greatest solve for xdle; right, so today with 499 I got "x-499;s largest prime divisor is 7". So, because I can't be bothered to find a suitable mid-point of the remaining candidates, I guess 729. 729 gives me "729 - x is prime", a hint which I've stated my displeasure of. But I'm slowly coming around to like it.

Now, at this point, the sensible thing to do would have been to guess 604 or 639 or something. But I wanted my 3-win dang it, so I didn't want to hazard a guess.

So instead I wrote out all 33 candidates (well, 32, since 506 is out). Here to we can rule out some of them; 729 doesn't give "largest prime divisor is 3", so we can't have anything divisible by 3. That's a third of them out. Also, 729 - x is prime, so any of the odd number candidates are also out. That's roughly half of the remaining candidates gone. That leaves us with 10, of which we can rule out a lot of them:

That leaves us with (548, 562, 646, 688). There's no way to find out which is the correct one, right?

Wrong. 729 - x is prime, which means that 729 + x is also prime, per my last comment. And so we've got:

• 729 + 646 = 1375 = 5*275
• 729 + 688 = 1417 = 13*109

So it all comes down to a coin toss. 548 or 562. I'll let you guess which one was the right answer (confession: I thought one of them was divisible by 17 because I didn't carry a 1, so I didn't actually realize it was a coin toss until I wrote this).

Right, but that requires memorizing squares up to at least 50, which I think is more unreasonable for most people.

IMO even going up to 25 is a bit much for most people, but yeah, I can see how 50 is really extreme.

From high school until I was almost 30, I only ever tried this up to about 600² and for numbers around 1000²,

Ah so I've got plenty of time to learn and catch up lol

I eliminated the 458 because of the likelihood of it giving "(499-x) is a prime number" clue

I mean I'd wager that there's a hint priority system, and it goes "guess-x or x-guess largest prime", then "guess+x largest prime", then "guess-x or x-guess is prime", so had 458 been the answer, the hint would've been "499+x's largest prime divisor is 29" (because 458 + 499 = 957 = 29 * 33).

Today's xdle is nothing to write home about. 499 gives "largest prime divisor is 101", which means that guessing any of the candidates gets you an easy 2 or 3; you can eliminate 600 based on what I've said above, which leaves only (701, 802, 903). 701 is the answer and is the easy 2. 802 gives x<802 which only leaves 701. 903 gives the same prime divisor hint, which eliminates 802 (and 903 obviously), which again only leaves 701. And even if you completely hypothetically did a dumb and guessed 600, 600 gives "x-600 is prime", which again eliminates the rest.

It make sense that (100n + y)² is easier to calculate, what with less overlapping additions.

I am a bit inconsistent when squaring numbers that end in 26 through 74 on whether I round to the nearest 50 vs. the nearest 100.

Well, I say that the inconsistency puts the mental in mental math.

Double the 3 is 6
Multiply 6 by 33 = 198

Would it be easier to double 33 and then multiply by 3? Though I dunno if 6*33 is harder than 3*66, and if at this point muscle memory makes you prefer the former.

However, being able to quickly do mental arithmetic is still helpful.  It gives you more information than you'd have otherwise.

Oh absolutely, I don't deny that, mental math generally faster than a phone, especially when you've perfected it to a T. As a tangent, it blows me away how so many people just... don't know their times tables. Like, even something as "simple" as 7*6 is apparently enough to make them reach for their phone calculators.

Unfortunately, I was careless and failed to recognize that 030 and 285 are both divisible by 5, so its largest divisor would have been 15.

It happens. Today (10/18) I had the misfortune of guessing 253 as my 2nd choice, when I should've gone for an even number (because since 253 gives me the same clue again, I can only narrow it down to (171, 120, 89, 48, 7)); guessing 294 gives (171, 89, 7), which is much more likely to guess correctly and get a 3-win (and also with 294 I can eliminate 171 since it and 294 are divisible by 3).

Feels like something I could do in my spare time, thanks for the strat :)

2500n² ± 100y + y²

The rest of the comment is correct, but you've forgotten the n for the center term (insert FOIL joke here).

3-digit squares shouldn't take more than a few seconds

That's really impressive!

Do you prefer squaring n then multiplying by 25, or is it better to do the squares that end in 5 (so, for 407, do you prefer 400*400 + 100*7*8 + 49 or 2500 * 64 + 100*7*8 + 49)?

 I'd first start learning up to about 125², which isn't hard and is impressive and useful enough.

As a newbie to squaring numbers, just memorizing the first 25 will be my first task lol

I wouldn't worry about doing 4-digit numbers, anyway.

I mean we have phones/calculators that can calculate squares for us. To me, memorizing squares/doing mental math is just a fun way to pass the time + a neat trick to be able to do.

On a somewhat related note, today's (10/17) xdle has "x+499 is square", which I would swear is not a coincidence. After a run of bad luck starting with 499, I finally understand how effective 499 is. Even luckier, we once again have a candidate (230) which is in the center of the list and divisible by 10, which I find aesthetically pleasing and gets me a three-guess win. Although I will (humorously) note that (512,243) is also an easy 3, but only because 512 gives a prime hint.

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.

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

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).

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)

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.

Having no 2 or 3 factors should be worse than a single 2 or 3.  Having no 2 factors leaves roughly half the numbers as candidates.  Being divisible by 2, but not 4, limits it to about a quarter.

Sure, but in the context of 512, not getting the largest divisor hint means that we can rule out half of the candidates of whatever other clue it gives us. But that's not something I ought to count on, obviously (today (10/9) 512 is unhelpful).

I've never seen the code for how xdle works and how it chooses its clues.  A strict priority list of what clues it prefers follows what I've seen.  I do know it's deterministic and not random as the same puzzle returns the same clues if the input is the same.

Not that it's likely to frequently be useful, but what do you think the priority list is? To me, it's probably "largest divisor" > "is prime/nth power" > "largest prime divisor" [ETA: Guessing 0 or 1 gives the amount of prime factors x has, but I'm pretty sure that that's only for 0 and 1]

for xdle, I only guess numbers that are possible solutions.

Seems reasonable enough. I'm probably more used to having a multitude of options (before I switched to 499 as my start). Speaking of,  if you guessed 701, you'd get "largest prime divisor is 19". Would you say it's better to guess 834 or 853? My intuition is 853 ('cause it's odd), but I'd like to know your thoughts.

That's fair enough. Thanks for the tips. I didn't have time for yesterday's xdle, but yeah, 14 candidates doesn't sound great. 786 is divisible by 2 and 3 only once, so that's a worst case scenario there using my usual strat.

Today's (10/8), however, is really funny to me, because 512 and 499 give the same clue, given that they are 13 apart from each other. The 19 or 20 candidates are both whittled down to the same 3.

However, if I'm understanding you correctly, it seems you're saying is that xdle has a priority list for the hints it gives. If so, then that's a big tell, because if I don't get "x and 512 largest divisor is 2", then I know that the number is odd. 512 would've thus been more useful here; 499 gives 19 candidates, and 512 gives 20, but the even numbers can be discarded without thought, and thus we only have 10 to choose from.

That doesn't mean that 512 is always better, and frankly, I wouldn't know how to most efficiently proceed (for today's xdle, my opening was (499, 243)) after getting that list. 243 seems good enough to split the list in half, although I'd be interested in knowing what you did.

Fair, though I will point out that my strategy is to use (512, 729) or (512, 243), which is only slightly worse than your strat (usually 10-ish possibilities are left), as opposed to being horrendously inefficient (yes, I realize I also sacrifice the potential to get consistent threes).

I will point out that, while the "largest divisor is X" clue is the worst clue in terms of narrowing down the answers, "is prime" is so unbearably annoying to list out the remaining possibilities (esp since I use only pen and paper) that it might as well have not narrowed it down at all.

After having tried 499 as the 1st guess for free play, I must say that I might start doing that instead. Though I do wonder how you narrow it down after entering 499?

Funny, I do the opposite. I pick powers of 2 and 3 because they're surprisingly useful.

I always start with 512 (2^9) because it will always half the possibilities, then 729 (3^6) or 243 (3^5), depending on whether x is greater or smaller than 512, because that also roughly splits the remaining possibilities in half.

Now, why the powers? Because if it gives me "x and 512's largest divisor is 2", then that tells me a lot; it has to be even, and it can't be divisible by 4, 8, 16, etc. (its largest divisor with 512 = 2^9 wouldn't have been 2 otherwise).

3 narrows it down even further, for the same reasons as above.

On top of that, I can multiply the numbers together; for example, if x's largest divisor with 512 is 2 and with 243 is 9, then x has to be divisible by 18 (otherwise, one of the conditions wouldn't be met).

Let's use October 5th's number as an example.

I wrote in 512, and got "x > 512; x and 512's largest divisor is 4". So I wrote in 729 and got "x and 729's largest divisor is 9", with x > 729. By multiplying, I get that x has to be divisible by 36. That means my possibilities are 756, 792, 828, 864, 900, 936, and 972. Further math shows that 792, 864, and 936 are divisible by 8, and 756 and 972 are divisible by 27, so they all have to go. That leaves me with just 828 and 900, which is just a toss-up.