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 #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 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.

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:

(please click 'Go !')

(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.

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The 206 solved problems had an average difficulty of 27.5% at Project Euler and
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.

more about me can be found on my homepage, especially in my coding blog.
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