Post by luvrhino in Working out process

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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.

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