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

# Problem 42: Coded triangle numbers

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

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 #ifdefs 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 result;
while (true)
{
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)
{
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.

Number of test cases (1-5):

Input data (separated by spaces or newlines):
Note: This live test requires Hackerrank-style input: enter a number and it will tell you its triangle number index or return -1

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
May 9, 2017 read words from STDIN

# Hackerrank

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 never 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.

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)
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.

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

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The 160 solved problems had an average difficulty of 21.8% at Project Euler and I scored 11,807 points (out of 13100) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
 << problem 41 - Pandigital prime Sub-string divisibility - problem 43 >>
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