<< 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?

# 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.

# 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.

# 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)*

# 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:

Python: 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)

# 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 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.

red problems are solved but exceed the time limit of one minute.

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

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I scored 12,626 points (out of 14300 possible points, top rank was 20 out ouf ≈60000 in July 2017) at Hackerrank's Project Euler+.

Look at my progress and performance pages to get more details.

My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

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