3 ms·
Interesting to try 140.something is what I'm guessing. Edit: Ah wait cube root. That is the square root I just tried. 26.something is what my mental calculati
by timwaagh 4y ago
Interesting to try 140.something is what I'm guessing.
Edit: Ah wait cube root. That is the square root I just tried.
26.something is what my mental calculations gave me. I thought it was going to be 27. Turns out it is. I made an error somewhere which is why you gotta do this on paper.
As for method I kind of brie brute forced it. started with 100 cubed which is hundred thousand then 20 cubed which is 8000 then 30 cubed which is 3 cubed times 10 cubed which is 27k. 25 cubed which I don't remember but was less then, then I knew it could not be 26 because the number is not even. I'd deal with fractions later. so I tried 27 cubed. 27 cubed is 3^3^3=2733^5=8133^4=24333^3= 72927=72920+7297
72920=72902=14400+180=14580
7297=4900+140+63=5103
Add them and 19683 comes out
Now try that in your head and it's a good recipe for frying one's brain.
- sp332 4y agoMuch too large. 100 cubed is 1,000,000.
- colinbeveridge 4y agoFor the square root, I get 140.3. My heuristic is to find a nearby square (here, 140^2 = 19600) and divide the discrepancy (83) by double the root of the nearby square (280). This is the number to adjust by. [^0] [^0] The Maclaurin series expansion of (A^2 + x)^(1/2) starts A + x/(2A), which is where this comes from. 83/280 is close to 84/280, which is 12/40 or 0.3. If I was trying to show off [^1], I might adjust that -- 1/28 is a quarter of a seventh, about 0.0357, so I can take 0.004 off of my adjustment to get 0.296 or 0.297. [^1] I mean, show off more. (The actual answer is 140.29611 -- the "0.3" adjustment is right to one part in 36,000; "0.296" is right to one part in 1.2 million.)