<< problem 274 - Divisibility Multipliers Linear Combinations of Semiprimes - problem 278 >>

# Problem 277: A Modified Collatz sequence

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:

Input data (separated by spaces or newlines):
Note: Enter the minimum number and the first characters of a Collatz sequence

This is equivalent to
echo "1000000 DdDddUUdDD" | ./277

Output:

(please click 'Go !')

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. Or just jump to my GitHub repository.

       #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%).

# Heatmap

Please click on a problem's number to open my solution to that problem:

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I stopped working on Project Euler problems around the time they released 617.
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The 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and
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.

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