<< problem 274 - Divisibility Multipliers | Linear Combinations of Semiprimes - problem 278 >> |
Problem 277: A Modified Collatz sequence
(see projecteuler.net/problem=277)
A modified Collatz sequence of integers is obtained from a starting value a_1 in the following way:
a_{n+1} = a_n/3 if an is divisible by 3. We shall denote this as a large downward step, "D".
a_{n+1} = (4a_n + 2)/3 if an divided by 3 gives a remainder of 1. We shall denote this as an upward step, "U".
a_{n+1} = (2a_n - 1)/3 if an divided by 3 gives a remainder of 2. We shall denote this as a small downward step, "d".
The sequence terminates when some a_n = 1.
Given any integer, we can list out the sequence of steps.
For instance if a_1=231, then the sequence {a_n}={231,77,51,17,11,7,10,14,9,3,1} corresponds to the steps "DdDddUUdDD".
Of course, there are other sequences that begin with that same sequence "DdDddUUdDD....".
For instance, if a_1=1004064, then the sequence is DdDddUUdDDDdUDUUUdDdUUDDDUdDD.
In fact, 1004064 is the smallest possible a_1 > 10^6 that begins with the sequence DdDddUUdDD.
What is the smallest a_1 > 10^15 that begins with the sequence "UDDDUdddDDUDDddDdDddDDUDDdUUDd"?
My Algorithm
The function isGood
returns true if its parameter x
has a Collatz sequence that starts with the characters found in parameter sequence
.
The loop in main
finds the first number whose Collatz sequence starts with "U", beginning at 10^15 (and finds a match 10^15).
In the next iteration, all potential candidates must be 3^1=3 apart.
10^15 doesn't match the two-character prefex "UD", but 10^15+3 does.
In the third iteration, all potential candidates must be 3^2=9 apart.
Again, 10^15+3 fails to match "UDD" but 10^15+9 matches.
Alternative Approaches
It's possible to solve this problem without a computer, too.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "1000000 DdDddUUdDD" | ./277
Output:
Note: the original problem's input 1000000000000000 UDDDUdddDDUDDddDdDddDDUDDdUUDd
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>
#include <string>
// return true if x matches the partial Collatz sequence
bool isGood(unsigned long long x, const std::string& sequence)
{
for (auto s : sequence)
{
switch (x % 3)
{
case 0: // large downward step
if (s != 'D')
return false; // failed
x /= 3;
break;
case 1: // upward step
if (s != 'U')
return false; // failed
x = (4*x + 2) / 3;
break;
case 2: // small downward step
if (s != 'd')
return false; // failed
x = (2*x - 1) / 3;
break;
default: ; // never reached
}
}
return true;
}
int main()
{
std::string sequence = "UDDDUdddDDUDDddDdDddDDUDDdUUDd";
auto current = 1000000000000000ULL;
std::cin >> current >> sequence;
// initially search every number (3^0=1), then every third (3^1), every ninth (3^2), ...
unsigned long long step = 1;
// look for a match of the first characters of sequence
for (size_t length = 1; length <= sequence.size(); length++)
{
// extract first characters
auto partial = sequence.substr(0, length);
// and find first match for those characters
unsigned int iterations = 0;
while (!isGood(current, partial))
{
current += step;
if (++iterations > 100) // quick hack: prevent invalid input of live test
return 1;
}
// increase step size for next iteration
step *= 3;
}
// found it
std::cout << current << std::endl;
return 0;
}
This solution contains 14 empty lines, 7 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
July 11, 2017 submitted solution
July 11, 2017 added comments
Difficulty
Project Euler ranks this problem at 35% (out of 100%).
Links
projecteuler.net/thread=277 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/LaurentMazare/ProjectEuler/blob/master/e277.py (written by Laurent Mazare)
Python github.com/Meng-Gen/ProjectEuler/blob/master/277.py (written by Meng-Gen Tsai)
Python github.com/smacke/project-euler/blob/master/python/277.py (written by Stephen Macke)
C++ github.com/roosephu/project-euler/blob/master/277.cpp (written by Yuping Luo)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem277.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem277.go (written by Frederick Robinson)
Mathematica github.com/steve98654/ProjectEuler/blob/master/277.nb
Perl github.com/shlomif/project-euler/blob/master/project-euler/277/euler-277-v2.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 274 - Divisibility Multipliers | Linear Combinations of Semiprimes - problem 278 >> |