For those of us who have tried but failed to memorize our logs (obviously the best method to ballpark it) :
For the n-the square root just group the digits by n starting from the ones. That will tell you how many digits you're dealing with. Simplish math will give you the first digit off of the last grouping (simple since you only have to ^n the numbers from 1-10)
Simple counting and a bit of math and I have the order of mag and first digit.
2_
At this point I can either think of a decent trick to iterate the second digit (quick Taylor? [1]), or just guesstimate a linear fit and call it a 5.
25 is within 10%.
[1] instead of Taylor I would do Pascal's triangle on the cube with the next term to calculate (30-x)^3 (I use 30 - x and not 20+x since I want x as small as possible and I know the answer is closer to 30 than 20.
Pascal's triangle:
1(30)^3 + 3(30)^2x + 330x^2 + 1x^3
Drop the higher two terms.
3^3 + 33^2x = 19
19-27 = 27x
X = 8/27 = 1/3 (- 1/27)
30-3.3 = 26.66
I use the sign of the next term of pascal's triangle to see if Ive under or over estimated
33*(-0.3)^2 -> positive
So my x is too large
So my answer is an underestimate.
(Cube root of 19000) greater than but close to 26.6
For the n-the square root just group the digits by n starting from the ones. That will tell you how many digits you're dealing with. Simplish math will give you the first digit off of the last grouping (simple since you only have to ^n the numbers from 1-10)
Simple counting and a bit of math and I have the order of mag and first digit.
2_
At this point I can either think of a decent trick to iterate the second digit (quick Taylor? [1]), or just guesstimate a linear fit and call it a 5.
25 is within 10%.
[1] instead of Taylor I would do Pascal's triangle on the cube with the next term to calculate (30-x)^3 (I use 30 - x and not 20+x since I want x as small as possible and I know the answer is closer to 30 than 20.
Pascal's triangle:
1(30)^3 + 3(30)^2x + 330x^2 + 1x^3
Drop the higher two terms.
3^3 + 33^2x = 19
19-27 = 27x
X = 8/27 = 1/3 (- 1/27)
30-3.3 = 26.66
I use the sign of the next term of pascal's triangle to see if Ive under or over estimated
33*(-0.3)^2 -> positive
So my x is too large
So my answer is an underestimate.
(Cube root of 19000) greater than but close to 26.6