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.