<< problem 13 - Large sum | Lattice paths - problem 15 >> |

# Problem 14: Longest Collatz sequence

(see projecteuler.net/problem=14)

The following iterative sequence is defined for the set of positive integers:

if n is even: n \to n/2

if n is odd: n \to 3n + 1

Using the rule above and starting with 13, we generate the following sequence:

13 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1

It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms.

Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.

Which starting number, under one million, produces the longest chain?

*NOTE:* Once the chain starts the terms are allowed to go above one million.

# Algorithm

The longest Collatz chain below five million contains 597 elements (and starts with 3732423).

A brute-force algorithm solves this problem within a half a second.

A smarter approach is to cache all chain lengths we encounter along the way.

My function `steps(x)`

tries to look up the result for `x`

in a `cache`

:

- if it succeeds, just return that cached value

- if it fails, we do one step in the Collatz sequence and call `step`

recursively

The code runs approx. 10x faster now (but needs about 10 MByte RAM more compared to the brute-force version).

## Alternative Approaches

There are surprisingly few "thresholds", where the Collatz chain length is longer than anything we have seen before:

` 1, 2, 3, 6, 7, 9, 18, 25, 27, 54,`

` 73, 97, 129, 171, 231, 313, 327, 649, 703, 871,`

` 1161, 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529,`

` 26623, 34239, 35655, 52527, 77031, 106239, 142587, 156159, 216367, 230631,`

` 410011, 511935, 626331, `

*837799*,1117065,1501353,1723519,2298025,3064033,3542887,

` 3732423`

This sequence is all we need to solve the original problem (well, this list contains all numbers up to 5000000 but we only need up to 1000000).

You can easily see that 837799 is the largest number less than one million.

## Modifications by HackerRank

I compute the Collatz sequences "on-demand":

if the input value `x`

exceeds `maxTested`

then all still unknown Collatz sequences up to `x`

will be analyzed.

Whenever a sequence is at least as long as the longest sequence we observed so far, then I have to update `longest`

.

This data structure `longest`

is supposed to hold the values shown above.

However, the modified Hackerrank problem asks for the *largest* number with a given Collatz chain length

but those numbers refers to the *smallest*.

Therefore I have to insert all numbers where

`length >= longest.rbegin()->second`

instead of

`length > longest.rbegin()->second`

In the end, my `std::map`

contains a few more values:

` 1, 2, 3, 6, 7, 9, 18, 19, 25, 27,`

` 54, 55, 73, 97, 129, 171, 231, 235, 313, 327,`

` 649, 654, 655, 667, 703, 871, 1161, 2223, 2322, 2323,`

` 2463, 2919, 3711, 6171, 10971, 13255, 17647, 17673, 23529, 26623,`

` 34239, 35497, 35655, 52527, 77031, 106239, 142587, 156159, 216367, 230631,`

` 410011, 511935, 626331, 837799,1117065,1126015,1501353,1564063,1723519,2298025,`

` 3064033,3542887,3732423`

## Note

I observed a max. recursion depth of 363 (incl. memoization). This shouldn't be a problem for most computers.

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <iostream>
#include <vector>
#include <map>
// memoize all paths length for n up to 5000000

const size_t MaxN = 5000000 + 2;
// we could change MaxN at will:
// it just affects performance, not the result

// identify chain lengths we haven't computed so far

const int Unknown = -1;
// store chain lengths

std::vector<short> cache(MaxN, Unknown);
// recursively count steps of Collatz sequence

unsigned short steps(unsigned long long x)
{
// finished recursion ?
if (x == 1)
return 1;
// try to use cached result
if (x < cache.size() && cache[x] != Unknown)
return cache[x];
// next step
long long next;
if (x % 2 == 0)
next = x / 2;
else
next = 3 * x + 1;
// deeper recursion
auto result = 1 + steps(next);
if (x < cache.size())
cache[x] = result;
return result;
}
int main()
{
// [smallest number] => [chain length]
std::map<unsigned int, unsigned int> longest;
// highest number analyzed so far
unsigned int maxTested = 1;
longest[maxTested] = 1; // obvious case
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int x;
std::cin >> x;
// compute remaining chain lengths
for (; maxTested <= x; maxTested++)
{
// get chain length
auto length = steps(maxTested);
// at least as long as anything we have seen before ?
if (length >= longest.rbegin()->second)
longest[maxTested] = length;
}
// find next longest chain for numbers bigger than x
auto best = longest.upper_bound(x);
// and go one step back
best--;
std::cout << best->first << std::endl;
}
return 0;
}

This solution contains 12 empty lines, 17 comments and 3 preprocessor commands.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

This is equivalent to`echo "1 5000000" | ./14`

Output:

*Note:* the original problem's input `1000000`

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)*

# Benchmark

The correct solution to the original Project Euler problem was found in **0.03** seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

Peak memory usage was about 12 MByte.

(compiled for x86_64 / Linux, GCC flags: `-O3 -march=native -fno-exceptions -fno-rtti -std=c++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

February 23, 2017 submitted solution

March 31, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler014

My code solved **13** out of **13** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **5%** (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 never an option.

# Links

projecteuler.net/thread=14 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/project-euler-14/ (written by Kristian Edlund)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p014.hs (written by Nayuki)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p014.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p014.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/10-19/problem14.c (written by eagletmt)

Go: github.com/frrad/project-euler/blob/master/golang/Problem014.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/14 Longest Collatz sequence.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler014.scala (written by Michael Bayne)

# Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.

yellow problems score less than 100% at Hackerrank (but still solve the original problem).

gray problems are already solved but I haven't published my solution yet.

blue problems are already solved and there wasn't a Hackerrank version of it (at the time I solved it) or I didn't care about it because it differed too much.

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

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