Post by luvrhino in Working out process

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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.

Kinwoods1 year ago

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).

luvrhino1 year ago

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.

Kinwoods1 year ago

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.

Kinwoods1 year ago

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:

Candidates	499 hint: 7 is x-499's largest prime divisor	729 hint: 729-x is prime
520		729 - 520 = 209 = 11*19. Not prime
548
562
590	590 - 499 = 7*13. 7 is not its largest prime divisor.
604		729 - 609 = 125 = 5*25. Not prime
632	632 - 499 = 7*19. 7 is not its largest prime divisor.
646
674		729 - 674 = 55 = 5*11. Not prime
688
716	716 - 499 = 7*31. 7 is not its largest prime divisor.

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).

luvrhino1 year ago

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.]

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