<< problem 161 - Triominoes | Cross-hatched triangles - problem 163 >> |
Problem 162: Hexadecimal numbers
(see projecteuler.net/problem=162)
In the hexadecimal number system numbers are represented using 16 different digits:
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
The hexadecimal number AF when written in the decimal number system equals 10*16+15=175.
In the 3-digit hexadecimal numbers 10A, 1A0, A10, and A01 the digits 0,1 and A are all present.
Like numbers written in base ten we write hexadecimal numbers without leading zeroes.
How many hexadecimal numbers containing at most sixteen hexadecimal digits exist with all of the digits 0,1, and A present at least once?
Give your answer as a hexadecimal number.
(A,B,C,D,E and F in upper case, without any leading or trailing code that marks the number as hexadecimal and without leading zeroes,
e.g. 1A3F and not: 1a3f and not 0x1a3f and not $1A3F and not #1A3F and not 0000001A3F)
My Algorithm
A nice dynamic programming problem ... my function count
has 5 parameters:
digits
stands for the number of digits, initially 16haveAny
is only true if any of the preceding digits is different from zerohaveZero
is only true if at least one of the preceding digits is0
haveOne
is only true if at least one of the preceding digits is1
haveA
is only true if at least one of the preceding digits isA
1. anything except
0
, 1
, A
:→ that means 13x everything with
digits - 1
→ since it's not
0
we have at least one digit different from zero, i.e. haveAny
must be true2. current digit is
0
:→ if there was already a zero, it's the same like case 1
→ if there was no zero so far, then a zero is only allowed if
haveAny
is true (no leading zero !)3. current digit is
1
:→ if there was already a one, it's the same like case 1
→ if there was no
1
so far, then set haveOne
to true, haveAny
as well4. current digit is
A
:→ exactly the same thinking like case 3 but set
haveA
to true instead of haveOne
I abort early if all conditions
haveZero
, haveOne
and haveA
are fulfilled.
Modifications by HackerRank
All results have to be modulo 10^9+7 and up to 100 digits are allowed.
The result should not be displayed in hexadecimal.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This live test is based on the Hackerrank problem.
This is equivalent toecho 3 | ./162
Output:
Note: the original problem's input 16
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).
//#define ORIGINAL
#include <iostream>
#include <iomanip>
// recursively count solutions
unsigned long long count(unsigned int digits, bool haveAny = false,
bool haveZero = false, bool haveOne = false, bool haveA = false)
{
// solved ?
if (haveZero && haveOne && haveA && digits < 15)
return 1ULL << (4 * digits); // same as pow(16, digits);
// processed all digits ? (but no combination of 0, 1, A found)
if (digits == 0)
return 0;
// assume current digit is not 0, 1 or A
unsigned long long next = count(digits - 1, true, haveZero, haveOne, haveA);
unsigned long long result = 13 * next;
// try to use a zero (only allowed if already have any non-zero digit => "no leading zero")
result += haveZero ? next : count(digits - 1, haveAny, haveAny, haveOne, haveA);
// try to use 1
result += haveOne ? next : count(digits - 1, true, haveZero, true, haveA);
// try to use A
result += haveA ? next : count(digits - 1, true, haveZero, haveOne, true);
#ifndef ORIGINAL
result %= 1000000007ULL;
#endif
return result;
}
int main()
{
#ifdef ORIGINAL
std::cout << std::uppercase << std::hex << count(16) << std::endl;
#else
unsigned int digits;
std::cin >> digits;
std::cout << count(digits) << std::endl;
#endif
return 0;
}
This solution contains 8 empty lines, 7 comments and 7 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
May 25, 2017 submitted solution
May 25, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler162
My code solves 11 out of 11 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 45% (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=162 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python blog.dreamshire.com/project-euler-162/
Python github.com/LaurentMazare/ProjectEuler/blob/master/e162.py (written by Laurent Mazare)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p162.py (written by Nayuki)
Python github.com/smacke/project-euler/blob/master/python/162.py (written by Stephen Macke)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/162.cc (written by Meng-Gen Tsai)
C++ github.com/roosephu/project-euler/blob/master/162.cpp (written by Yuping Luo)
Java github.com/HaochenLiu/My-Project-Euler/blob/master/162.java (written by Haochen Liu)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p162.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem162.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem162.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p162.mathematica (written by Nayuki)
Mathematica github.com/steve98654/ProjectEuler/blob/master/162.nb
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p162.hs (written by Nayuki)
Perl github.com/shlomif/project-euler/blob/master/project-euler/162/euler-162.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 161 - Triominoes | Cross-hatched triangles - problem 163 >> |