<< problem 144 - Investigating multiple reflections of a laser ... | Investigating a Prime Pattern - problem 146 >> |
Problem 145: How many reversible numbers are there below one-billion?
(see projecteuler.net/problem=145)
Some positive integers n have the property that the sum n + reverse(n) consists entirely of odd (decimal) digits.
For instance, 36 + 63 = 99 and 409 + 904 = 1313. We will call such numbers reversible; so 36, 63, 409, and 904 are reversible.
Leading zeroes are not allowed in either n or reverse(n).
There are 120 reversible numbers below one-thousand.
How many reversible numbers are there below one-billion (10^9)?
My Algorithm
A simple and straightforward brute force solution - and it finishes in less than a second.
I'm pretty sure there is a smarter approach but from time to time you need to unleash the raw power of modern CPUs ...
Actually, I have added two important optimization:
- there are no solutions between 10^8 and 10^9 → makes the program 10x faster
- either the first or the last digit must be odd → assume the last is odd and multiply result by 2 → 2x faster
Interactive test
This feature is not available for the current problem.
My code
… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.
#include <iostream>
#include <vector>
// note: my code only reversed odd numbers, they can't produce a leading zero
unsigned int reverse(unsigned int x)
{
unsigned int result = 0;
while (x > 9) // speed optimization: skip the last digit
{
auto digit = x % 10;
result *= 10;
result += digit;
x /= 10;
}
// simple processing of the last digit, saves one division/modulo ;-)
return result * 10 + x;
}
// return true if all digits are odd
bool onlyOdd(unsigned int x)
{
while (x > 0)
{
// found an even digit ?
if (x % 2 == 0)
return false;
x /= 10;
}
// yes, all odd
return true;
}
int main()
{
// if the first digit is odd, then the last has to be even
// the same logic applies in reverse order, too
// therefore analyze only odd numbers and multiply the result by 2
const unsigned int factor = 2;
unsigned int count = 0;
// no solutions between 10^8 and 10^9
for (unsigned int i = 11; i < 100*1000*1000; i += factor)
if (onlyOdd(i + reverse(i)))
count += factor;
std::cout << count << std::endl;
return 0;
}
This solution contains 7 empty lines, 9 comments and 2 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.8 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
May 23, 2017 submitted solution
May 23, 2017 added comments
Difficulty
Project Euler ranks this problem at 20% (out of 100%).
Links
projecteuler.net/thread=145 - 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-145-how-many-reversible-numbers-are-there-below-one-billion/ (written by Kristian Edlund)
C# github.com/HaochenLiu/My-Project-Euler/blob/master/145.cs (written by Haochen Liu)
Python blog.dreamshire.com/project-euler-145/
Python github.com/hughdbrown/Project-Euler/blob/master/euler-145.py (written by Hugh Brown)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p145.py (written by Nayuki)
Python github.com/sefakilic/euler/blob/master/python/euler145.py (written by Sefa Kilic)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/145.cc (written by Meng-Gen Tsai)
C++ github.com/roosephu/project-euler/blob/master/145.cpp (written by Yuping Luo)
C++ github.com/smacke/project-euler/blob/master/cpp/145.cpp (written by Stephen Macke)
C++ github.com/zmwangx/Project-Euler/blob/master/145/145.cpp (written by Zhiming Wang)
C github.com/LaurentMazare/ProjectEuler/blob/master/e145.c (written by Laurent Mazare)
Java github.com/dcrousso/ProjectEuler/blob/master/PE145.java (written by Devin Rousso)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p145.java (written by Nayuki)
Go github.com/frrad/project-euler/blob/master/golang/Problem145.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p145.mathematica (written by Nayuki)
Mathematica github.com/steve98654/ProjectEuler/blob/master/145.nb
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p145.hs (written by Nayuki)
Perl github.com/gustafe/projecteuler/blob/master/145-reversible-digits.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 144 - Investigating multiple reflections of a laser ... | Investigating a Prime Pattern - problem 146 >> |