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After a few boring days (e.g., yesterday's, if starting with 499 and the perfect cube clue, should have taken exactly 3 guesses), today's was a testament to reverting to my original strategy of maximizing factors on your second guess.

• x<499; (499 - x)'s largest prime divisor is 17

My initial instinct was to use 15 * 17 = 255 since that splits the two halves most evenly.  However, 499 - 255 = 244 = 2² * 61.  That's only 6 factors.

However, if I subtracted 289 instead, my guess would be 210 = 2 * 3 * 5 * 7 for 16 factors.  That was too tempting to pass up.  This paid off as my clue was "x<210; x and 210's largest divisor is 5."  That narrows it down to one candidate.  5 * 17 is 85, so 210 - 85 = 125.  However, 210 - 170 = 40, would not be as then the largest divisor would have been 10.

Even if that had gone the other direct and was "x>210;  x and 210's largest divisor is 5," then the answer would have been 210 + 85 = 295, since 210 + 255 = 465 would have had the largest divisor of 15.

Had I gone with the more median choice of 244, I would have received the less useful repeat of the "(244-x)'s largest prime divisor is 17" clue.  Unless the return clue from 244 was "largest divisor is 4," "largest divisor is 61," or a random clue of "(244 x)'s largest divisor is <something big>," then the clue from 210 was going to be more useful.

I firmly believe that 210 was the best option for the second guess.

Similarly, had the first clue gone the other direction with "x>499; (x - 499)'s largest prime divisor is 17," the best second guess would be 720 (2⁴ * 3² * 5 for 30 factors) and not the more central 754.
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ETA (11/2): Today had the same scenario with "x<499; (499-x)'s largest prime divisor is 71."  Assuming 499-71 is impossible, there are 6 candidates: 499 - 71n, where n is 2, 3, 4, 5, 6. or 7.  The median is n = 4.5.

Using n=4 yields a guess of 215 = 5 * 43.  Won't yield any "x and 210's largest divisor is 5" clues as adding or subtracting 355 would put it out of range.  215 risks a repeat of "(499-x)'s largest prime divisor is 71."

Meanwhile, n=5 yields a guess of 144, which is fantastic.  That has a lot more factors than we can use.  Regardless of the direction, it could give a response of x and 144's largest divisor is 2, 3, or 4....though it shouldn't be 4.  That would allow you to nail the next guess.  Otherwise, if the solution were 144 ± 71, then you'd either get a "is prime" or a different clue that you could pinpoint the answer.

It turns out that using n=4 was fine as instead of (x-215) is prime, it gave "(x+215)'s largest prime divisor is 167," which also points to 286.

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

I wanted 8 or 16 for my third guess to add one or two more factors of 2 beyond just being divisible by 4.  That way, if my next clue still said divisible by 4, I'd eliminate around half the candidates.  If it said 8, but my guess was divisible by 16, that would divide the candidates by 4.

Hey, same here! Today I tried (499, 744). 499 gives "largest prime divisor is 7", and 744 gives "x-744 is a square".

You made this more complicated than necessary given that you involved Wolfram Alpha.  I also started (499, 744).  At that point, (x-744) had to be a square number that's divisible by 7.  That left only 7² and 14² as 21² is too big.  Unfortunately, neither 793 - 499 = 294 nor 940 - 499 = 441 have a prime factor larger than 7, which makes it a coinflip.

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))

I ended up getting lucky based on somewhat bad thinking.  For my initial guess, I was just going to split the prime numbers under 499 in half.  I ended up overestimating how biased those prime numbers were toward lower numbers.  I estimated the median prime candidate would be 191 or 193.  It turns out that it's 223 (48th prime of 95 under 500).  191 is the 43rd.  

I went with 308 because it was "good enough" both in terms of being towards (what I thought was) the median and 308 = 2² * 7 * 11 having 12 factors.

x<308; x and 308's largest divisor is 4.

There were still too many candidates left for me to bother listing them.  Since I was now looking at prime numbers between 191 & 499, I knew those were fairly evenly distributed.  My goals at this point were to find a number where (499-x) was prime and I wanted many non-7 and non-11 factors.  I wanted it to be divisible by 8, if not 16.

I started at 152 and worked my way up:

• 152 = 8 * 19.  347 is prime, but I'm only adding another 2 and a 19 as prime factors.
• 160 - 339 isn't prime.
• 168 = 8 * 3  * 7.  Divisible by 7, not a candidate.
• 176 - 323 isn't prime.
• 184 - 315 isn't prime.
• 192 = 2⁶ * 3.  307 is prime.  14 factors is decent. 

Good enough.  Quickly check under 152:

• 144 - 355 isn't prime.
  
• 136 - 363 isn't prime.

Based on that, I went with 192 and got lucky.

Had I gone 499 -> 243 ("x<243; x and 243's largest divisor is 3") my next guess would have been 120.  120 splits the remaining numbers evenly, has more factors (16) than any number smaller than it.  379 is prime.  120 generates the "largest divisor is 24," which is lovely.  

144 and 216 aren't valid because they're divisible by 9.   240 isn't valid because it's divisible by 120.  168 and 192 are the only options left.  499 + 168 = 667 = 23 * 29.  That seems like it'd generate a "(x+499)'s largest prime divisor is 29" clue.

Meanwhile, 499 + 192 = 691, which is prime.

For that reason, between those two candidates, I'd pick 192.

[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.]

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!)

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'm not sure how many people are playing.

I'll keep my eyes open for your theory.  I didn't get the impression that's always the case, but I may have missed it.

> 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.

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.  So, since 744 was 499 + 35 * 7, I needed numbers that were 499 + n * 7, where n was 33, 29, 25, 21, 17, 13, 9, 5, or 1.  When n was 33, that makes x = 730, which is divisible by 2 and not 4.

Put another way, starting at 744, the next smaller number that is divisible by 2 and where (x - 499) is divisible by 7, is 744 - 14 = 730.  From that point, the numbers that have that property occur at intervals of 28...702, 674, etc..  

Mentally, I didn't think of it as n mod 4 = 1.  Instead, I saw that n = 33 was my starting point and started decrementing by 4s, eliminating any values of n that had a prime factor more than 7.

I thought I had responded to your previous message, but I can't find it anywhere.

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

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.

As for 10/21, I continued my policy of only using possible solutions.  After 499, I tried 499 + 7 * 35 = 744, which is divisible by 24.  My return clue was "x<744; x and 744's largest divisor is 2."  That means my solution isn't divisible by 3 or 4.  Which means it is also 499 +n * 7 where n mod 4 = 1, and n has no prime factors above 7.

So, n is one of: 1, 9, 21, or 25.  n can't be 5, because 534 is divisible by 3.  Meanwhile, n can't be 13, 17, 21, 29, nor 33 because then 7 wouldn't be the largest prime divisor of x - 499.  The candidates are then: 506, 562, 646, 674:

• 506 is unlikely because it might have generated "x - 499 is prime" as its first clue.
• 562 + 744 = 1306 = 2 * 653.  That clue seems fine.  653 is probably too big for "fairly large."
• 646 + 744  = 1390.  Were 646 the answer, I think the second clue would be "x + 744's largest divisor is 139."
• 674 + 744 = 1418 = 2 * 709.  That clue seems fine.

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

Right, but that requires memorizing squares up to at least 50, which I think is more unreasonable for most people. That's why I explained it with the 50n.  I strictly used the 50n for quite a while except for numbers near between 410 & 590 or 910 and higher.  From high school until I was almost 30, I only ever tried this up to about 600² and for numbers around 1000²,  except for rare occasions.  It was only then that I realized expanding to 4-digit, 5-digit, and larger numbers was quite feasible.

> Double the 3 is 6 Multiply 6 by 33 = 198. 
Would it be easier to double 33 and then multiply by 3?

You can.  In my case, I automatically double the first term before really thinking about what the y is.  I see that my first term is going to be 300 or 350 and instantly know that means we're going up by 600s or 700s.  There may be exceptions to this if the y-term ends in 5, but normally it's an automatic step that I don't even think about.

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).

With the largest prime divisor of 41 clue, there are 12 candidates below 499.  I eliminated the 458 because of the likelihood of it giving "(499-x) is a prime number" clue and I chose the median of the remaining 11 candidates, 212, which is divisible by 4.  Like you, I received the same "largest prime factor is 41" clue again.  However, that allowed me to eliminate more candidates because your smallest factor was 11.

212 was 499 - (7 * 41).  Since it didn't give me the "x and 212's largest divisor is 2 or 4" clue, that left 171, 089, and 007.  I would lean against 171 because I didn't have "(212-x) is a prime number" and for lack of anything else, I chose the middle one of the three to guarantee I'd get it in 3 or 4.

As for the guesses that were 41 away, only one gave the prime number clue:

• x>48; (x-48) is prime
• x<130; (x+130)'s largest prime divisor is 73

So, yes, you can't rule anything out just because you didn't get "(x - y) is prime" clue.  You can use that to lean toward one number vs. the other like with 212 vs. 253 with today's puzzle.  212 being divisible by 4 while 253 being an unhelpful 11 * 23 is a more important reason to do that, as you explained above.

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.

You did better than I.

Since 25² and 27² are a 4th power and a 6th power, respectively, I eliminated those as candidates.  That left the squares of 23, 24, 26, 28, 29, 30, and 31.  I went with 28², which was the middle.  285 gave me "x < 285; x and 285's largest divisor is 3."  24² would be 077, which isn't divisible by 3.  Both 030 and 177 are.  

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.

I guessed 030, when it clearly had to be 177.  I didn't even notice I messed that up until right now.  I thought I just lost a coinflip.

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)?

That's an excellent question.  That formula was more for proof concept (i.e., eliminating carries and making the units and tens digit controlled only by the y² term).  In practice, I don't pay attention to what the n is nor think of the formula.

Mentally, what I'd do for 407²:

1. Double the 4 is 8
2. Multiply that by 7.  That's 56.
3. 40² is 1600
4. Add them: 1656
5. Multiply by 100: 165,600
6. Put the 7² on the end.
7. 165,649

I do the second term first, because that's the variable and most complicated term...and by complicated, I mean 2 * 4 * 7.

Ones where the first term ends in 50 are more tricky, because the 350² has four-digits instead of just two.  That means I'm going to have to do real addition or subtraction.  For 333²:

1. (350 - 17)²
2. Double the 3.5 is 7
3. Multiply 7 by 17 = 119
4. 35² is 1225
5. 1225 - 119 = 1106
6. Multiply by 100 = 110,600
7. Add 17² (289).
8. 110,889

Step 5 annoys me,  So, taking advantage of the fact that, for practical purposes, I have all two-digit squares memorized, I would do calculate this as (300 + 33)².  Rounding to the nearest 100 usually causes the middle term have more digits  However, the first term now only affects the ten-thousands and hundred-thousands digit, making the adding or subtracting even easier:

1. (300 + 33)²
2. Double the 3 is 6
3. Multiply 6 by 33 = 198
4. 30² is 900
5. 900 + 198 = 1098
6. Multiply by 100 = 109,800
7. Add 33² (1089).
8. 110,889

Step 5 is now trivial, though the addition in step 7 is likely to be more complicated.  It will be an addition and an addition of at most two digits...in this case, ignoring the tens and units digits, the addition is just 1098 + 10.  Using 350², I had to subtract a 3-digit number from a 4-digit number.  The good thing there is that I'm extremely familiar with the 4-digit number and that number will always end in 25.

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.  However, I never use the first term as 450, 550, 950, or 1050.  That's because using 500 and 1000 are even easier, because the middle term will have a 0 as its hundreds digit:

(500 ± y)² = 
250,000 ± 2 * 500 * y + y² =
250 * 1000 ± y * 1000 + y² =
(250 ± y) * 1000 + y² =

That is super easy.  I just add or subtract y from 250.  That's the thousands.  Then add y².  

For 576² -> 250 + 76 = 326 -> 326,000 + 76².

76² = 5776 -> 576² = 331,776

And, in practice, rather than add or subtract y from 250, I take the original 576 and subtract 250 from it.  It equals, 326 either way.

There are several other optimizations from having done this multiple times, but I've already complicated it more than is ideal.  This may be something to bookmark or save somewhere and revisit to incrementally develop your ability, if you want to do it at all.

Even if you don't have squares up to 25² memorized, you could use this for calculating squares between 40² and 60² or 90² and 110² at first and build from there.

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.

It is a stupid human trick and, yes, it would be more useful in the years before people carried computers in their pockets everywhere.  However, being able to quickly do mental arithmetic is still helpful.  It gives you more information than you'd have otherwise.  In an extremely dorky hypothetical, it can allow you to quickly spot when your local Which Wich sub shop is calculating 8.25% incorrectly on your $5.75 sub.  You can then inform them and their district office and they'll give you free subs.  

Hypothetically.

I had the exact same route to 3 guesses on today's puzzle.  Very nice.

Square Numbers

Okay, strap in.  I don't know the exact number I have memorized at this point, but it'd be probably in somewhere in the 80s before I'd have a do a quick sanity check to make sure.  However, I can square rather large numbers in my head, though 5- and 6-digit ones are usually slow.  3-digit squares shouldn't take more than a few seconds.

I'll cut-and-paste and adapt stuff I've written elsewhere on how to do it.  I'm confident that you could at least mentally calculate at least three-digit squares fairly quickly without too much practice.  My method is one I developed while bored in high school English class years ago.  Others have certainly independently come up with it.

First, you need to memorize squares up to 25, but that's it.

For squares larger than 25, I rewrite them in the format:

(50n ± y)², where y is between 0 and 25.

(50n ± y)² =
2500n² ± 2 * 50 * y + y² =
2500n² ± 100y + y²

This means the last two-digits are only affected by the y² term.  The other two terms are hundreds and above.  Moreover, the y² term only contributes a number between 0 & 6 to the hundreds digit, which is easy to handle.

For example,

64² =
(50 + 14)² =
50² + 2 * 50 * 14 + 14² =
2500 + 14 * 100 + 14² =
3900 + 14² =
3900 + 196 =
4096

283² =
(300 - 17)² =
300² - 2 * 300 * 17 + 17² =
90,000 - (6 * 17) * 100 + 17² =
100 * (900 - 102) + 17² =
100 * (798) + 17² =
79,800 + 289 =
80,089

Now, this does involve knowing what all the squares ending in -50 (i.e., 100n + 50)...350, 450 , 550, etc.

I've got a trick for that, too.  To square a number ending in 5 (10n + 5), all you need to do is:

1) Remove the ending 5.
2) Take the remaining number and multiply it by one plus that number (n(n+1)).
3) Take that result and postpend 25 on the end.  That's the solution.

So;

35² = 1225
45² = 2025
55² = 3025
65² = 4225
95² = 9025
2835² = 8,037,225

For that last one, I used the result above for 283² (80,089), added 283 (80,372), and then put a 25 on the end.

Now, if you want to calculate 763²:

(750 + 13)² =
(75²)(10²) + 2 * 750 * 13 + 13² =
5625 * 100 + 15 * 13 * 100 + 13² =
(5625 + 195) * 100 + 13² =
582,000 + 169 =
582,169

For 4-, 5-, and 6-digit numbers, instead of 50n, I'd use 500n, 5000n, 50000n, respectively.  That means the range of y goes up by a factor of 10 each time, between 0 & 250, 0 & 2500, etc.  .However, that means I have to repeat the process multiple times:

Leveraging the answer for 283² = 80,089, let's calculate 4283²:

(4000 + 283)² =
4000² + 2 * 4000 * 283 + 283² = 
16,000 * 1000 + 8 * 283 1000 * + 283² =
(16,000 + 2264) * 1000 + 283² =
18,264 * 1000 + 283² =
18,264,000 + 80,089 =
18,344,089

Please note in all these calculations how easy all the addition steps are.  There is usually only one or two non-zero digits that overlap.  The 769² one was the hardest because I had 5625 + 195, which still isn't that bad.

I wouldn't worry about doing 4-digit numbers, anyway.  I'd first start learning up to about 125², which isn't hard and is impressive and useful enough.  Your next goal might be up to around 625².  That way, the middle term in the expansion only involves multiplying a number from 1 thru 25 by a number 1 thru 12:

(550 + 3 )² =
302,500 + 2 * 550 * 3 + 3² =
3025 * 100 + (11 * 3) * 100 + 3² =
(3025 + 33) * 100 + 3² =
305,800 + 9 =
305,809

This can expand as far as you need.  The problem is that once you get to 5- and 6-digits, you need to repeat the process multiple times and memory becomes a problem:

123,456² =
(100,000 + 23,456)²
10 billion + 2 * 100,000 * 23,456 + 23,456²

Which means I now have to do (20,000 + 3456)², which involves 456², which is (500 - 44)².  There's no reason to need to do that for numbers that large unless you get bored easily.  Calculating it in this iterative manner, starting with 456², minimizes how much short-term memory I need.

Finally, being able to square numbers quickly is useful because then you can often use the difference of two squares to multiply two different numbers very fast. 

24 * 36 =
(30 - 6)(30 + 6) =
30² - 6² =
900 - 36 =
864

Leveraging the calculation of 283² from above:

274 * 292 =
(283 - 9)(283 + 9) =
283² - 9² =
80,089 - 81 =
80,008

Because I'm so adept with calculating square numbers, I use difference of two squares a lot.  This quite a bit to digest.  I'm obviously open to clarify any of that.

xdle BONUS:  As part of that y² term being the only part that affects the last two digits, that means all 22 possible two-digit endings are found in 1² through 24².  24² and 26² share the same two-digit ending (576 & 676).  Same with 23² and 27² (529 & 729).  Same with 1², 49², 51², 99², 101², etc.   

Moreover, the numbers 2, 3, 7, and 8 will never be in the units digit.  That can help you quickly judge some of those "(x + 792) is a perfect square" type clues.

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.

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:

1. 702 mod 11 = 9
2. (2 * 702) mod 11 = 7, since 18 mod 11 = 7
3. This means the first candidate is 702 -7 = 695.
4. My guess will be 695 - 11n, for some value of n that cuts the remaining numbers in half...so I want n around 31.
5. (702 + 695) = 1397 = 127 * 11.
6. If (x + 702) were 90 * 11, that would be easy and fulfill the largest prime divisor being 11.
7. 127 - 90 = 37
8. 695 - 11 * 37 = 
9. 695 - 407 =
10. 288
11. 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
12. (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.

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.

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.

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.

This is still an issue 15 months later.  I've played for months and never noticed it before today because it should only happen if my initial guess is 1 away from the solution. I can refresh the page and make a new guess in the place of the one that is one away.  It should be an easy enough to debug.  

My best guess is that it gets into an infinite loop when trying to discover what clue to give next.  More specifically, since it prioritizes clues like "(x - 500) is a 6th power," it may get into a loop trying to figure out what the highest power is that works.  Since 1^n = 1, there's no limit to what power it could be.

The solution is to just special case this scenario and, perhaps, just give away the answer by saying the guess is one away.

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.

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.

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.

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.

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).

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.

Just so I don't sound too smart, unlike you, I only went to Heardle after reading DanHoelck's comment noting it was a prank.  That was a massive hint. 

The way they executed it on the page confirmed I was correct so I didn't even bother to click the link.

Heh.  I love that before I even clicked on the link, I correctly predicted what the Heardle song would be from your comment.

Granted, I've been on the internet during the past decade.  Many people would have predicted the same.

I fortunately had the first three letters and had eliminated all other vowels, so I was able to type combos quickly until I got one that wasn't in red.  That was my guess of the most likely thing to be a word with what I had left after five guesses, but yeesh.

This is the first time I haven't known a word in Wordle, Dordle, or Quordle.  I've been playing since Jan 1 (Wordle) or mid-Jan for the other two.

I wasn't anticipating an April 1 joke, but I agree that's the most likely answer.  It's not using random words from word lists, since otherwise there would be regular plurals.

You should be able to just tap and drag down (as though you were trying to scroll up) and then release.  Alternatively, if you click the three dots in the upper left corner to open the menu, there is a clockwise arrow/circle that you can click to refresh.

You shouldn't need to clear cookies or anything complicated.  The developer suggested that just changing colors would be sufficient to reset the word list.  

Try refreshing the page on both by pressing <F5>.

The developer updated the word lists, but you need to refresh the browser for the lists to be updated.  

https://zaratustra.itch.io/dordle/devlog/347485/more-words-less-colors

Try refreshing the page on both by pressing <F5>.

The developer updated the word lists, but you need to refresh the browser for that to matter.  Having the Daily Dordle change based on browser wouldn't make sense:

https://zaratustra.itch.io/dordle/devlog/347485/more-words-less-colors