<< problem 260 - Stone Game | Pseudo Square Root - problem 266 >> |
Problem 265: Binary Circles
(see projecteuler.net/problem=265)
2^N binary digits can be placed in a circle so that all the N-digit clockwise subsequences are distinct.
For N=3, two such circular arrangements are possible, ignoring rotations:
For the first arrangement, the 3-digit subsequences, in clockwise order, are:
000, 001, 010, 101, 011, 111, 110 and 100.
Each circular arrangement can be encoded as a number by concatenating the binary digits starting with the subsequence of all zeros
as the most significant bits and proceeding clockwise. The two arrangements for N=3 are thus represented as 23 and 29:
00010111_2 = 23
00011101_2 = 29
Calling S(N) the sum of the unique numeric representations, we can see that S(3) = 23 + 29 = 52.
Find S(5).
My Algorithm
I wrote a straightforward backtracking solution that start with N bits which are all zero and calls search
recursively:
- try to append another zero, try to append another one
- if the most recent N bits occurred in the sequence before, then go back
- if the sequence has 2^N bits, then another solution was found
history
tracks all numbers.If its k-th bit is used, then the number k is already part of the sequence.
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 | ./265
Output:
Note: the original problem's input 5
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 <iostream>
// N = number of bits
unsigned int bits = 5;
// find all valid results
unsigned long long search(unsigned long long history, unsigned long long sequence)
{
// bitmask for all relevant bits
const unsigned int Mask = (1 << bits) - 1;
// seen all 2^bits bit combinations ?
unsigned long long allBits = (1ULL << (1 << bits)) - 1;
if (history == allBits)
{
// remove lowest bits (which are all zeros)
sequence >>= bits - 1;
return sequence;
}
// shift most recent bits one position to the left, remove highest bit
unsigned int next = (sequence << 1) & Mask;
// two options for the following value
unsigned int zero = next;
unsigned int one = next + 1;
// sum of all found sequences
unsigned long long result = 0;
// check "zero"
if ((history & (1ULL << zero)) == 0)
result += search(history | (1ULL << zero), sequence << 1);
// check "one"
if ((history & (1ULL << one )) == 0)
result += search(history | (1ULL << one ), (sequence << 1) | 1);
return result;
}
int main()
{
std::cin >> bits;
std::cout << search(1, 0) << std::endl;
return 0;
}
This solution contains 10 empty lines, 10 comments and 1 preprocessor command.
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
July 1, 2017 submitted solution
July 1, 2017 added comments
Difficulty
Project Euler ranks this problem at 40% (out of 100%).
Links
projecteuler.net/thread=265 - 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/p265.py (written by Nayuki)
Python github.com/smacke/project-euler/blob/master/python/265.py (written by Stephen Macke)
Python github.com/steve98654/ProjectEuler/blob/master/265.py
C++ github.com/Meng-Gen/ProjectEuler/blob/master/265.cc (written by Meng-Gen Tsai)
C++ github.com/roosephu/project-euler/blob/master/265.cpp (written by Yuping Luo)
C github.com/LaurentMazare/ProjectEuler/blob/master/e265.c (written by Laurent Mazare)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p265.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem265.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem265.go (written by Frederick Robinson)
Mathematica github.com/steve98654/ProjectEuler/blob/master/265.nb
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p265.hs (written by Nayuki)
Perl github.com/shlomif/project-euler/blob/master/project-euler/265/euler-265-v1.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 260 - Stone Game | Pseudo Square Root - problem 266 >> |