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Problem 139: Pythagorean tiles
(see projecteuler.net/problem=139)
Let (a, b, c) represent the three sides of a right angle triangle with integral length sides.
It is possible to place four such triangles together to form a square with length c.
For example, (3, 4, 5) triangles can be placed together to form a 5 by 5 square with a 1 by 1 hole in the middle
and it can be seen that the 5 by 5 square can be tiled with twenty-five 1 by 1 squares.
However, if (5, 12, 13) triangles were used then the hole would measure 7 by 7 and these could not be used to tile the 13 by 13 square.
Given that the perimeter of the right triangle is less than one-hundred million, how many Pythagorean triangles would allow such a tiling to take place?
My Algorithm
The sides of each triangle are a <= b < c.
In this picture shown above, c is the length of the "outer square" and b - a is the length of the "hole".
A tiling exists only if c is a multiple of b - a, that means c % (b - a) == 0
.
For the (3,4,5) triangle: c mod (b-a) = 5 mod (4-3) = 5 mod 1 = 0 → yes, tiling exists
For the (5,12,13) triangle: c mod (b-a) = 13 mod (12-5) = 13 mod 7 = 6 → no
I generate all basic Pythagorean triplets (see problem 75) and check if c % (b - a) == 0
.
Then all multiples create a valid tiling, too: k * c mod (k * b - k * a) = k * c mod k * (b - a) = k * (c mod (b - a)) and if x mod y = 0 then always kx mod y = 0.
There are \lfloor dfrac{10^8}{a+b+c} \rfloor valid multiples for each basic triplets.
Note
There is a subtle problem with the way basic triplets are generated: a and b are not always a <= b ... sometimes a > b.
So actually the correct test for a valid tiling is c % abs(b - a) == 0
(that's what my diff
variable is for).
When I looked at the valid basic triplets then I saw that b - a = 1 in all cases. There are actually only 10 basic triplets.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "1 12" | ./139
Output:
Note: the original problem's input 100000000
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)
My code
… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.
#include <iostream>
// find greatest common divisor
template <typename T>
T gcd(T a, T b)
{
while (a != 0)
{
T c = a;
a = b % a;
b = c;
}
return b;
}
int main()
{
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
unsigned int maxLength = 100000000;
std::cin >> maxLength;
unsigned int numValid = 0;
for (unsigned int m = 2; 2*m*m < maxLength; m++)
for (unsigned int n = 1; n < m; n++)
{
// only valid m and n
if ((m + n) % 2 != 1)
continue;
if (gcd(m, n) != 1)
continue;
// compute basic triplet
auto a = m*m - n*n;
auto b = 2*m*n;
auto c = m*m + n*n;
auto sum = a + b + c;
// length of a side of the hole
unsigned int diff = b - a;
if (a > b)
diff = a - b;
// check if outer square is a multiple
if (diff > 0 && c % diff == 0)
{
// number of valid triplets derived from the current basic triplet
auto moreTriples = maxLength / sum;
numValid += moreTriples;
}
}
std::cout << numValid << std::endl;
}
return 0;
}
This solution contains 8 empty lines, 6 comments and 1 preprocessor command.
Benchmark
The correct solution to the original Project Euler problem was found in 0.8 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++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
June 1, 2017 submitted solution
June 1, 2017 added comments
Difficulty
Project Euler ranks this problem at 50% (out of 100%).
Links
projecteuler.net/thread=139 - 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-139-pythagorean-tiles/ (written by Kristian Edlund)
C# github.com/HaochenLiu/My-Project-Euler/blob/master/139.cs (written by Haochen Liu)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p139.py (written by Nayuki)
Python github.com/smacke/project-euler/blob/master/python/139.py (written by Stephen Macke)
Python github.com/steve98654/ProjectEuler/blob/master/139.py
C++ github.com/Meng-Gen/ProjectEuler/blob/master/139.cc (written by Meng-Gen Tsai)
C++ github.com/roosephu/project-euler/blob/master/139.cpp (written by Yuping Luo)
C++ github.com/steve98654/ProjectEuler/blob/master/139.cpp
Java github.com/dcrousso/ProjectEuler/blob/master/PE139.java (written by Devin Rousso)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p139.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem139.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem139.go (written by Frederick Robinson)
Mathematica github.com/steve98654/ProjectEuler/blob/master/139.nb
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p139.hs (written by Nayuki)
Perl github.com/shlomif/project-euler/blob/master/project-euler/139/euler-139-v1.pl (written by Shlomi Fish)
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 and/or exceed time/memory limits.
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 | |
black | problems are solved but access to the solution is blocked for a few days until the next problem is published | |
[new] | the flashing problem is the one I solved most recently |
I stopped working on Project Euler problems around the time they released 617.
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I scored 13526 points (out of 15700 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 !!!
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