Post by luvrhino in Working out process

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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.

Because I thought 646 was unlikely, I guessed 562 on the theory that it splits 506 and the other two candidates. Guessing 674 is less good because it's not one of the middle two.

Kinwoods1 year ago (1 edit)

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!)

luvrhino1 year ago

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.

Kinwoods1 year ago

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))

luvrhino1 year ago (4 edits)

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.]

Kinwoods1 year ago

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...

luvrhino1 year ago

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.

Kinwoods1 year ago

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".

luvrhino1 year ago (1 edit)

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.