I took a class with the Sanjoy (the author) this past fall. Awesome guy and a great teacher.
Street-Fighting Math: Educated Guessing and Opportunistic Problem Solving (2010) [pdf]
31–32 of 32 posts
Re: Street-Fighting Math: Educated Guessing and Opportunistic Problem Solving (2010) [pdf]
#32Earlier quoted context omitted.
pi^2=10 so pi^6=1000? This is within 5% of the real answer, but does not not involve music!
I chose a coincidentally easy problem (whoops). Here's another one, less accessible to attack: Find e^7. Or 25000^(1/8). The trick to using the musical scale is that it's logarithmic in frequency but linear in key (ABCDEFG). The frequency step between adjacent notes is 2^(1/12); the frequency jump over an octave is 2; the frequency jump over 40 notes is 10. There are other handy ratios to remember, like that going up…
A good trick for calculating powers of e is that e^3 ~ 20. So e^7 is roughly 400e, so about 1080, completely without pen and paper. This is only 1.5% off.
Your trick is cool too, by the way.
For your other challenge, I would use the enormously useful 2^10 = 1024 ~ 1000 (which the musical trick implicitly uses too, in the 40 notes is 10 times the frequency mnemonic). So 25000 is close to 5^2 * 2^10, and it's square root is 5 * 2^5 = 160 (we can always keep in mind that 2^10 ~ 1000 is an overestimate by 2.4%, so our 160 ~ sqrt(25000) is an overestimate by 1.2%, so an improved approximation of sqrt(25000) is 158). 158 is very close to the midpoint of 144 and 169, so its square root is not far from the midpoint of 12 and 13. So my approximation of 25000^(1/4) is 12.5. Finally I need to take a sqrt of that. I know the square of 3.2 is 10.24 (our old friend 2^10 = 1024), and that's about 20% too low. Ok, we need to bump our 3.2 up by 10%, that makes is 3.52, and that's our answer. It's only 0.7% off, but it's only by luck. For a more general problem, keeping a tab of your first estimate as an absolute number and your current correction as a percentage gives an error of less than 2%.