<< problem 39 - Integer right triangles | Pandigital prime - problem 41 >> |

# Problem 40: Champernowne's constant

(see projecteuler.net/problem=40)

An irrational decimal fraction is created by concatenating the positive integers:

0.12345678910\red{1}112131415161718192021...

It can be seen that the 12th digit of the fractional part is 1.

If d_n represents the nth digit of the fractional part, find the value of the following expression.

d_1 * d_10 * d_100 * d_1000 * d_10000 * d_100000 * d_1000000

# Algorithm

The original problem can be solved in a trivial way:

- a `for`

-loop appends numbers to a long string until that string contains enough digits

- read relevant digits, convert them from ASCII to integers and multiply them

The Hackerrank problem asks for digits at positions up to 2^18 which cannot be done the brute force way

because we would be running out of memory (and CPU time).

My function `getDigit`

finds a digit without building such a long string.

It is based on the following observation:

- there are 9 numbers with one digit (1 to 9)

- there are 90 numbers with one digit (10 to 99)

- there are 900 numbers with one digit (100 to 999)

- ... and so on

The first part of `getDigit`

figures out how many digits the number has which is pointed to by the parameter `pos`

.

- the first 9 numbers are represented by 1*9 digits in Champernowne's constant

- the next 90 numbers are represented by 2*90=180 digits

- the next 900 numbers are represented by 3*900=2700 digits

- ... and so on: `range`

will be 9, 90, 900, ... and `digits`

will be 1, 2, 3, ...

The variable `skip`

will contain 9, 9+2*90 = 189, 9+2*90+3*900 = 2890 until the next step would exceed `pos`

.

Now that the function knows how many digit the number (pointed to by `pos`

) has, `getDigit`

process its digits.

To do so, it moves `first`

closer to `pos`

by repeated adding `range`

.

Whenever `range`

becomes too large, the next (smaller) digit has to be adjusted until we have the final value of `first`

.

That number is converted to a string and the desired digit returned.

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <string>
#include <iostream>
// return the digit at position "pos"
// first digit after then decimal dot has pos = 1 (not zero !)

unsigned int getDigit(unsigned long long pos)
{
// assume pos has one digit
unsigned int digits = 1;
// then there are 9 other numbers
unsigned long long range = 9;
// the smallest of them is 1
unsigned long long first = 1;
// there are 9 numbers with 1 digit
// there are 90 numbers with 2 digits
// there are 900 numbers with 3 digits
// there are 9000 numbers with 4 digits
// ...
// let's figure out the number of digits
// skip numbers with too few digits
unsigned long long skip = 0;
while (skip + digits*range < pos)
{
skip += digits*range;
// digits = 2 => range = 90 and
// digits = 3 => range = 900
// digits = 4 => range = 9000, etc.
digits++;
range *= 10;
first *= 10;
}
// now that we know the number of digits
// adjust "first" and "skip" such that the left-most/highest digit of pos and skip are identical
// then continue with the next digit
while (range > 9)
{
// could be replace by some modular arithmetic, but I'm too lazy for tough thinking ;-)
while (skip + digits*range < pos)
{
skip += digits*range;
first += range;
}
// next lower digit
range /= 10;
}
// right-most digit (basically same inner loop as above when range == 1)
while (skip + digits < pos)
{
first++;
skip += digits;
}
// skip all "skippable" digits
pos -= skip;
// strings are zero-based whereas input is one-based
pos--;
// create a string version of our number
auto s = std::to_string(first);
// extract digit and convert from ASCII to an integer
return s[pos] - '0';
}
int main()
{
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int product = 1;
// read 7 positions
for (unsigned int i = 0; i < 7; i++)
{
unsigned long long pos;
std::cin >> pos;
// multiply all digits
product *= getDigit(pos);
}
// print result
std::cout << product << std::endl;
}
return 0;
}

This solution contains 12 empty lines, 28 comments and 2 preprocessor commands.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

This is equivalent to`echo "1 1 2 3 4 5 6 7" | ./40`

Output:

*Note:* the original problem's input `1 10 100 1000 10000 100000 1000000`

__cannot__ be entered

because just copying results is a soft skill reserved for idiots.

*(this interactive test is still under development, computations will be aborted after one second)*

# Benchmark

The correct solution to the original Project Euler problem was found in **less than 0.01** seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

(compiled for x86_64 / Linux, GCC flags: `-O3 -march=native -fno-exceptions -fno-rtti -std=c++11 -DORIGINAL`

)

See here for a comparison of all solutions.

**Note:** interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without `-DORIGINAL`

.

# Changelog

February 25, 2017 submitted solution

April 18, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler040

My code solved **9** out of **9** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **5%** (out of 100%).

Hackerrank describes this problem as **medium**.

*Note:*

Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.

In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is never an option.

# Links

projecteuler.net/thread=40 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/project-euler-40-digit-fractional-part-irrational-number/ (written by Kristian Edlund)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p040.hs (written by Nayuki)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p040.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p040.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/40-49/problem40.c (written by eagletmt)

Javascript: github.com/dsernst/ProjectEuler/blob/master/40 Champernowne's constant.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler040.scala (written by Michael Bayne)

# Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.

yellow problems score less than 100% at Hackerrank (but still solve the original problem).

gray problems are already solved but I haven't published my solution yet.

blue problems are already solved and there wasn't a Hackerrank version of it (at the time I solved it) or I didn't care about it because it differed too much.

*Please click on a problem's number to open my solution to that problem:*

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