<< 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
My 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
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
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, ... anddigits
will be 1, 2, 3, ...
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.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "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)
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.
Benchmark
The correct solution to the original Project Euler problem was found in less than 0.01 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++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 solves 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 rarely 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:
C# www.mathblog.dk/project-euler-40-digit-fractional-part-irrational-number/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/40-49/problem40.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p040.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/40 Champernowne's constant.js (written by David Ernst)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p040.mathematica (written by Nayuki)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p040.hs (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler040.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/040-Champerdownes-constant.pl (written by Gustaf Erikson)
Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own.
Maybe not all linked resources produce the correct result and/or exceed time/memory limits.
Heatmap
Please click on a problem's number to open my solution to that problem:
green | solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too | |
yellow | solutions score less than 100% at Hackerrank (but still solve the original problem easily) | |
gray | problems are already solved but I haven't published my solution yet | |
blue | solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much | |
orange | problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte | |
red | problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too | |
black | problems are solved but access to the solution is blocked for a few days until the next problem is published | |
[new] | the flashing problem is the one I solved most recently |
I stopped working on Project Euler problems around the time they released 617.
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I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
Look at my progress and performance pages to get more details.
Copyright
I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.
All of my solutions can be used for any purpose and I am in no way liable for any damages caused.
You can even remove my name and claim it's yours. But then you shall burn in hell.
The problems and most of the problems' images were created by Project Euler.
Thanks for all their endless effort !!!
<< problem 39 - Integer right triangles | Pandigital prime - problem 41 >> |