<< problem 41 - Pandigital prime | Sub-string divisibility - problem 43 >> |

# Problem 42: Coded triangle numbers

(see projecteuler.net/problem=42)

The nth term of the sequence of triangle numbers is given by, t_n = frac{n(n+1)}{2}; so the first ten triangle numbers are:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...

By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value.

For example, the word value for SKY is 19 + 11 + 25 = 55 = t_10. If the word value is a triangle number then we shall call the word a triangle word.

Using words.txt (right click and 'Save Link/Target As...'), a 16K text file containing nearly two-thousand common English words, how many are triangle words?

# My Algorithm

My function `getTriangle`

returns either `NoTriangle = -1`

or the triangle index if the parameter `x`

is a triangle number.

E.g. `getTriangle(55) = 10`

.

I use my own approximation formula for the triangle index which can be derived as follows:

A triangle number t_n is a defined as

t_n = n (n + 1) / 2

Ff x is such a triangle number t_n then

x = n (n + 1) / 2

2x = n (n + 1)

2x = n^2 + n

For any a^2 we know for a's successor

(a + 1)^2 = a^2 + 2a + 1

Therefore the relationship:

n^2 < n^2 + n < n^2 + 2n + 1

n^2 < 2x < (n + 1)^2

n < sqrt{2x} < (n + 1)^2

In order to find the triangle index n, I just compute n = sqrt{2x}.

If t_n = n (n + 1) / 2 = x then n is indeed what we were looking for - else the function returns `-1`

.

## Modifications by HackerRank

I just have to check numbers whether they are triangle numbers or not.

The major problem was that input values can be up to 10^18.

## Note

Solving the quadratic equation leads to the same result:

t_n = x

x = n (n + 1) / 2

x = (n^2 + n) / 2

2x = n^2 + n

0 = n^2 + n - 2x

n must be positive so only one solution is left:

n = frac{sqrt{1 + 8x} - 1}{2}

In the end you have to check this result as well: only if it an integer then n is our result.

Project Euler's file can be easily parsed in C++.

Initially I included it in my source code (which works flawlessly) but then decided to read from STDIN.

# 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 to`echo "1 55" | ./42`

Output:

*(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, as well as the input data, too.

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).

#include <string>
#include <iostream>
#include <cmath>
const int NoTriangle = 0;
// return triangle index or -1 if not a triangle number

int getTriangle(unsigned long long x)
{
unsigned long long n = sqrt(2*x);
// if n it truely the right answer then t(n) = x
unsigned long long check = n * (n + 1) / 2;
if (x == check)
return n;
else
return NoTriangle;
}
// read a single word from STDIN, syntax: "abc","def","xyz"

std::string readWord()
{
std::string result;
while (true)
{
// read one character
char c = std::cin.get();
// no more input ?
if (!std::cin)
break;
// ignore quotes
if (c == '"')
continue;
// finish when a comma appears
if (c == ',')
break;
// nope, just an ordinary letter (no further checks whether c in 'A'..'Z')
result += c;
}
return result;
}
int main()
{
//#define ORIGINAL

#ifdef ORIGINAL
unsigned int triangleWords = 0;
while (true)
{
// read next word
auto word = readWord();
if (word.empty())
break;
unsigned int sum = 0;
// A = 1, B = 2, ...
for (auto c : word)
sum += c - 'A' + 1; // all words contain only uppercase letters without spaces or other characters
// another "triangle word" ?
if (getTriangle(sum) != NoTriangle)
triangleWords++;
}
std::cout << triangleWords << std::endl;
#else
unsigned int tests;
std::cin >> tests;
while (tests--)
{
// all work is done in getTriangle()
unsigned long long x;
std::cin >> x;
std::cout << getTriangle(x) << std::endl;
}
#endif
return 0;
}

This solution contains 12 empty lines, 13 comments and 6 preprocessor commands.

# Benchmark

The correct solution to the original Project Euler problem was found in less than 0.01 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

(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 27, 2017 submitted solution

April 19, 2017 added comments

May 9, 2017 read words from STDIN

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler042

My code solves **7** out of **7** 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 rarely an option.

# Similar problems at Project Euler

Problem 45: Triangular, pentagonal, and hexagonal

*Note:* I'm not even close to solving all problems at Project Euler. Chances are that similar problems do exist and I just haven't looked at them.

# Links

projecteuler.net/thread=42 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

C#: www.mathblog.dk/project-euler-42-triangle-words/ (written by Kristian Edlund)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p042.hs (written by Nayuki)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p042.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p042.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/40-49/problem42.c (written by eagletmt)

Scala: github.com/samskivert/euler-scala/blob/master/Euler042.scala (written by Michael Bayne)

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.

# 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 |

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I scored 13,386 points (out of 15600 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 41 - Pandigital prime | Sub-string divisibility - problem 43 >> |