I think the key would be to figure out what combinations of guess and solution trigger what clues. You ideally want to trigger clues that narrow it down to one solution (e.g., (x + 499) is a perfect 10th power). I'm not sure how to do that. Obviously, the perfect 10th power clue is just sheer luck.
When I first started, I thought having an initial guess with the most possible factors was optimal. Thus, I started with 540 and then switched to 420 to capture the 7 as a factor. This turns out to be the exact opposite of what I want. The problem with having all those factors is that it triggers the "x and 540's largest divisor is 2" clue, which is a problem. That doesn't narrow down the list of solutions much at all...especially when subsequent guesses yield the exact same clue. My couple losses all came from that situation where I kept getting that same clue, and it became a matter of luck.
Instead, I found using 499 as a starter works much better. 499 is prime. The only time I received the "x and 499's largest divisor" clue was when the solution was 998. I tend to get much more useful clues this way.
Aside from that, it usually amounts to a binary search where I try to split the possible candidates in half...half lower than my guess and half higher. Note that this is candidates and not the raw number.
If I get the clue "x > 499; (x-499) is a perfect square," I do not guess 755, which would most closely divide the remaining range in half. Instead, I know that (x - 499) is between 1² and 22², so I'd rather choose 12² = 144 -> 643 as my next guess. There are 10 candidates above 643 and 11 below. In reality, I probably would guess 668 or 695, on the theory that for something like 515 or 563 it would be more likely to say (x - 499) was a perfect 4th power or cube, than perfect square.
I solve these in my head because I'm arrogant. It rarely causes problems, but today I had "(x + 767)'s largest prime divisor is 163," where a calculator would have made things slightly faster.
If others have insights on how to trigger better clues, I'd love to hear them.
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.
I agree that, instinctively, it makes sense to have as many factors as you want. That's precisely what I did by using 540 and 420, both of which have 24 factors. In practice, those tend to generate "largest divisor is X" clues, which are the worst you can get.
By starting with 499, I often get a clue like "(499 - x)'s largest prime divisor is 59." That is much more useful as it limits it to 8 solutions right away. Meanwhile, with your example of "x > 512; x and 512's largest divisor is 4," you still have 61 possible solutions. Obviously, 8 is much better.
I think for the Oct 5 puzzle, I had "(x -499)'s largest prime divisor is 47" as my first clue, which left 10 candidates, which is still much better than 61.
[ETA: I'm pretty sure with your "x > 512; x and 512's largest divisor is 4" clue, the answer will not be 512 plus a square number. That means you could eliminate +4, +36, +100, +196, +324, and +484...which brings it down to 55 candidates. I think you'd get a "(x - 512) is a perfect square" as your clue were the solution 516, 548, etc.
Similarly, it probably isn't 644 or 932, since that would generate a "(x + 512) is a perfect square" clue, which brings it down to 53 candidates. I don't know this for a fact, but xdle seems to generate these clues when possible.]
I encourage you to experiment with the free xdle using different starters, including 499. Take note of the number of possible solutions that are left after that initial number.
My stats for xdle and Dordle were reset a few weeks ago. Since then, in 23 games, I have zero losses, 3 twos, 14 threes, 3 fours, 2 fives, and 1 six. I doubt that's possible starting with 512.
October 6's puzzle is a bad example because it took me 4 guesses using 499 and 3 using 512 (pretending I didn't know the answer). Both numbers generate the kind of clue I was discussing, but 512 generates a better one (3 candidates for 512 vs. 8 candidates for 499).
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?
It's all pretty much a binary search of trying to divide the candidates in half with half lower and half higher. I occasionally tweak that when I think certain candidates are more or less likely or that if I use a certain guess I will prompt a (yyy - X) is a square number or (yyy - X) is prime clue. That's the same logic of why we start around 500.
Again, it's important that it's dividing the candidates in two and not the remaining numbers. I describe that above when I discuss what I do with "x > 499; (x-499) is a perfect square," Guessing 755 (i.e., 499 + 16²) would be the closest to the midpoint of 499 and 999. The remaining candidates are 499 + 1² through 499 + 22². That suggests using 11². However, because I think 4², 8², 9², 16², and possibly 1² would all have generated different clues (e.g., "(x - 499) is a perfect 6th power" for 563 = 499 + 8²), I think 13² is probably best as it divides the remaining choices in evenly...8 above and 8 lower.
13² has a secret bonus of possibly triggering "x < 668; (668 - x) is a perfect square" if the solution were 524 or 643, thanks to the Pythagorean theorem and the 5-12-13 triangle. Now that I think of it, 12² is even better for this since you'd have a 5-12-13, 9-12-15, and 12-16-20 triangles all as options. "x > 643; (x -643) is a perfect square" for 668 or 724 and "x > 643; (x -643) is a perfect 8th power" for 899.
While I (obviously) tend to overthink things, this was the first time I noticed those 12² options. I rarely anticipate the "(x + yyy) is a perfect square" type clues, though I love it when I get them.
Similarly, I don't necessarily go through listing out all the candidates after the first clue, especially if I start with a mediocre-to-bad clue like "x > 499; (x-499)'s largest prime divisor is 5." In that case, I'll just guess where the midpoint among candidates will be, use something like 679 (= 499 + 2² * 3² * 5) and hope to get a different clue next. I wouldn't be surprised if 659 or even 619 were better, here, but I can't be bothered.
I hope that makes sense as to what my thought process is on subsequent guesses and what kinds of things one could consider if you wanted to play optimally. I do try to strike a balance in how much time to spend to try playing optimally vs. getting the thing done more quickly.
Today (10/7) is one where using 499 is much better than your originally proposed method. I managed to get it in two. Starting with (512, 729) leaves 14 candidates, which isn't great. Depending on the clue, it is possibly better than if it didn't have a single 3 factor, only a single 2.
I think at that point using the 8th biggest would make sense, which is 870 assuming I'm counting correctly. That helps because if it returns with the largest divisor is 30, you'd know it's 750 or 930. Otherwise, since it's "x < 870; largest divisor is 6" you know it's not 750, leaving 6 candidates.
Using the 4th biggest, it's 822, which would leave 3 candidates. That is fine with two guesses remaining as you guess the middle one and go above or below if that's not correct. Instead, I got the "x<822; (822-x) is a square." Clearly (822 - x) = 6² -> x = 786...which was the middle of the 3 remaining, anyway.
With the benefit of hindsight, after the 3rd guess, I should have eliminated 834 because would have given the "is a square" clue instead of "largest divisor is 6." That would have left 5 candidates: 762, 786, 798, 822, 858. Choosing the middle one, means you'd either get it correct, get the "is a square clue" if it's 762, or get the "largest divisor is 6," if the solution is 786, 822, or 858. If it were 822 or 858 you're left with a coinflip to see if you get it in 5 or 6 guesses.
By contrast, with 499, my clue was "(x-499)'s largest prime divisor is 41." That leaves 11 or 12 candidates, since 540 might have generated a less useful (x-499) is prime clue. The 6th of the 11 or 7th of the 12 candidates is 499 + 7 * 41 = 499 + 287 = 786, which is the solution. That's luck, but it narrowed it down further in one guess than what 512/729 did in 2.
Anyway, I enjoy this kind of post-mortem analysis. It's fun to figure out what the decision tree should have been.
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.
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.
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. I am not aware of it not giving a "perfect square/cube/4th power...etc." clue when that's an option.
As for the 10/8 puzzle, I decided 20 * 13 was good enough, and got it in two clues. That was pure luck. I would have done the same if I started at 512. In that case, it switches to "x<252; (252-x) is prime," narrowing it down to just 239.
Getting the "(252-x) is prime" clue is what I'd expect, based on what I think the clue priority list is. Similarly, if I guess 303 today, it gives "(303-x) is a 6th power."
Understanding that there is this hierarchy of clues can help eliminate some clues. When I bother to look deeper as I have above, projecting what sequence of clues it would give one or two steps down the line can reduce the number of guesses.
Finally, I'll note that, unlike Dordle, for xdle, I only guess numbers that are possible solutions. I wouldn't do your (512, 243) scheme unless it fit. I don't make impossible guesses frequently in Dordle, but sometimes they're necessary to eliminate several letters at once.
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.
As I mentioned, I don't have it fully mapped out.
I think it puts (x - 499) is a perfect nth power as the highest priority. The other possibility is (x+499) is a perfect nth power. I haven't been able to test which one is higher since those would only both be available for the same number for specific values of x.
Using some prime numbers away from 910, I did get these clues just now:
(x+927)'s largest prime divisor is 167
(x+893)'s largest prime divisor is 601
(x+933)'s largest prime divisor is 97
(x+953)'s largest prime divisor is 23
It preferred those to (963-x) is prime. 910 + 963 = 1873 is a prime number. Same if I tried 957 as 1867 is prime.
I don't recall it not giving "x is a multiple of (499 - x)" when that is an option and it can't do the nth-power clue. It does that for 912, 913, 920, 923, 975, and other number today. It does prefer that to (499 - x) is prime clue.
My only other discovery just now is that if I am 1 away from the solution, the game breaks. Hitting <enter> will cause it to become unresponsive. I can reload the game and make a new guess, however. I will report that bug in a different thread. I don't think I had ever been one away before, so I don't know if this bug is new or not.