It’s time for another Project Euler exercise. This time around it’s problem 16:
215 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.
What is the sum of the digits of the number 21000?
I have tried to found an elegant way to solve this problem at a mathematical level, but I couldn't gain any insight relating powers of 2 to their representation in base 10 (I'd love to know if such a thing exists). Therefore I resorted to solve this problem with brute force.
Luckily F# (and now all other .NET 4.0 cousin languages) sports the
BigInteger numeric type which helps tackling such huge numbers.
There are basically two approaches to solve this problem:
- Transform the result of exponentiation into a string and iterate through the string adding up all the individual digits, or
- Do without strings, and decompose the big number into the individual digits on my own.
I have decided to take the second route because it seemed like a more interesting exercise (the first approach could probably lead to a one-liner). .
Let’s consider a generic number n and consider how we can describe it as a sum of powers of 10:
and k = 2, d0 = 3, d1 = 2, and d2 = 4
So, given a generic number n, we need to find the highest power of ten ( k ) such that
(logarithms will help us there). After k is found, we can start diving n by 10k to find dk and use the reminder of the division to calculate dk-1 and so forth until we reach d0.
#light open System open System.Numerics let rec decompose_rec n power_of_ten accu = match power_of_ten with | i when i = 1I -> accu @ [n] |_ -> let d = n / power_of_ten let digits = accu @ [d] decompose_rec (n - (d * power_of_ten)) (power_of_ten / 10I) digits let decompose n = let power = floor (log10 (float n)) let initial_power_of_ten = BigInteger.Pow(10I, int32 power) decompose_rec n initial_power_of_ten  let sum_digits n = decompose n |> List.sum
-  A recruiter for a very big company once asked me something very similar during an interview as a coding exercise ↩