<< problem 163 - Cross-hatched triangles | Intersections - problem 165 >> |
Problem 164: Numbers for which no three consecutive digits have a sum greater than a given value
(see projecteuler.net/problem=164)
How many 20 digit numbers n (without any leading zero) exist such that no three consecutive digits of n have a sum greater than 9?
My Algorithm
A straight-forward recursive solution:
- start with an "empty" number (no digits at all)
- append a digit (0 to 9) if the previous two digits and the current digits don't exceed 9
- stop if enough digits where appended (
digits = 0
)
cache
prevents re-computing the same results over and over again.
Modifications by HackerRank
There can be up to 100 digits.
Note
You can change the maximum sum of three consecutive digits, too (default is 9).
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "3 9" | ./164
Output:
Note: the original problem's input 20 9
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. Or just jump to my GitHub repository.
The code contains #ifdef
s to switch between the original problem and the Hackerrank version.
Enable #ifdef ORIGINAL
to produce the result for the original problem (default setting for most problems).
#include <iostream>
#define ORIGINAL
// Hackerrank: up to 100 digits
// Project Euler: 20 digits
const unsigned int MaxDigits = 100;
// bonus feature: change maximum sum of three consecutive digits
unsigned int maxSum = 9;
// count matching numbers
// prevprev: the digit two positions to the left of the current digit
// prev : the digit one position to the left of the current digit
// digits : number of remainings digits
// isFirst : true only for the first digit (which must not be zero)
unsigned long long search(unsigned int prevprev, unsigned int prev, unsigned int digits, bool isFirst)
{
// done ?
if (digits == 0)
return 1;
// memoize
unsigned int id = digits * 100 + prevprev * 10 + prev; // simple hash
static unsigned long long cache[(MaxDigits + 1) * 100] = { 0 };
if (cache[id] != 0)
return cache[id];
// iterate over all digits such that current + prev + prevprev <= 9
unsigned long long result = 0;
for (unsigned int current = 0; current + prev + prevprev <= maxSum; current++)
{
// no leading zero
if (current == 0 && isFirst)
continue;
// add next digit
result += search(prev, current, digits - 1, false);
}
#ifndef ORIGINAL
// Hackerrank only
result %= 1000000007;
#endif
// memoize
cache[id] = result;
return result;
}
int main()
{
unsigned int digits = 20;
std::cin >> digits;
if (digits > MaxDigits)
return 1;
#ifdef ORIGINAL
// bonus feature: change maximum sum of three consecutive digits
std::cin >> maxSum;
#endif
std::cout << search(0, 0, digits, true) << std::endl;
return 0;
}
This solution contains 13 empty lines, 16 comments and 6 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
June 16, 2017 submitted solution
June 16, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler164
My code solves 9 out of 9 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 45% (out of 100%).
Hackerrank describes this problem as easy.
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=164 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p164.py (written by Nayuki)
Python github.com/smacke/project-euler/blob/master/python/164.py (written by Stephen Macke)
Python github.com/steve98654/ProjectEuler/blob/master/164.py
C++ github.com/Meng-Gen/ProjectEuler/blob/master/164.cc (written by Meng-Gen Tsai)
C++ github.com/roosephu/project-euler/blob/master/164.cpp (written by Yuping Luo)
C++ github.com/steve98654/ProjectEuler/blob/master/164.cpp
C github.com/LaurentMazare/ProjectEuler/blob/master/e164.c (written by Laurent Mazare)
Java github.com/HaochenLiu/My-Project-Euler/blob/master/164.java (written by Haochen Liu)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p164.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem164.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem164.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p164.mathematica (written by Nayuki)
Mathematica github.com/steve98654/ProjectEuler/blob/master/164.nb
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p164.hs (written by Nayuki)
Perl github.com/shlomif/project-euler/blob/master/project-euler/164/euler-164.pl (written by Shlomi Fish)
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 163 - Cross-hatched triangles | Intersections - problem 165 >> |