Determine the smallest perfect square whose base ten representation begins with 2011 (reading left to right). How about the smallest positive perfect cube whose base ten representation begins with 2011 (reading left to right)?
*** Calculators are allowed, but no computer programs.
sqrt(2011) = 44.84417464955732
sqrt(2012) = 44.85532298401161
so if the answer has an even number of digits, its square root's digits must lie, lexicographically, above the first sequence of digits but below the second: 4485. The result is 4485^2 = 20115225.
But the answer might have an odd number of digits:
sqrt(20110) = 141.8097316829843
sqrt(20120) = 141.8449858119772
14181^2 = 201100761
The evendigit (8digit) one is smaller, so the answer to part 1 is 20115225.
For cube roots we have to try numbers of digits that are congruent to 1, 2 or 0, mod 3:
2011^(1/3) = 12.6222668331058
2012^(1/3) = 12.6243586904251
20110^(1/3) = 27.1938495320821
20120^(1/3) = 27.1983563020572
201100^(1/3) = 58.5873727874249
201200^(1/3) = 58.5970823289993
The three candidates are therefore 12623^3, 27194^3 and 5859^3. Clearly this last is the smallest: 5859^3 = 201127054779, which is thus the answer to part 2.
Edited on September 16, 2011, 1:54 pm

Posted by Charlie
on 20110916 13:48:37 