If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is A solution can be implemented quickly and intuitively by using an iterative approach that loops through a range of integers between 1 and Using the mod operator to check for even divisibility a zero remainder after division we sum those integers, ithat are divisible by 3 or 5.
The program runs instantly for upper bounds likebut does not scale well for larger ones such as 10 9. So, we need to find a more efficient way of calculating this sum without looping.
A formula attributed to Carl Friedrich Gauss will calculate the sum of the first n natural numbers. This is an example of a closed—form expression describing a summation. Write the numbers in two rows that wrap around as shown below:.
The sum of each column is 11 i. If n is odd, simply start with zero instead of one. Remember, when there is an odd number of elements we start from zero to keep the columns paired. In our Python function, sumn shown belowthis is accomplished by taking the floor of n divided by d to find the number of non—zero terms.
Then, calculate the sum using an expanded formula which accounts for the multiplier, d. This is a typical application of the inclusion—exclusion principle.
In general, sum the numbers less than that are divisible by 3 3, 6, 9, 12, 15… or 5 5, 10, 15… and subtract those divisible 3 and 5 15, 30, 45, …. This solution is much faster than using brute force which requires loops. Also note that we subtract one from the upper bound as to exclude it. HackerRank increases the upper bound from 1, to 1 billion and runs 10, test cases. The summation formula is the legacy of Carl Friedrich Gauss, the German mathematician. The sequence [1, 3, 6, 10, 15, …] is called the triangular numbers and count objects arranged in an equilateral triangle.
The game of bowling, or ten—pin, sets 10 pins in a equilateral triangular form: one pin in the first row through 4 pins in the last row. If you want to find the th triangular number, you begin the long and laborious addition of the first numbers. While the other students labored away, the ten—year—old Gauss handed his teacher the tablet with his answer within seconds. The teacher was surprised when he looked at the tablet to find the correct answer — 5, — with no steps in the calculation.
The teacher thought that Gauss must have cheated somehow. Rather than tackling the problem head on, Gauss had thought geometrically. He argued that the best way to discover how many beans there were in a triangle with rows was to take a second similar triangle of beans which could be placed upside down and adjacent to the first triangle.
Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. I've recently been working on Project Euler problems in Python. I am fairly new to Python, and still somewhat new as a programmer. In any case, I've ran into a speed-related issue coding a solution for problem 5.
The problem is. What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20? I've checked around some, and I haven't been able to find anything on this problem pertaining to Python specifically. There were some completed scripts, but I want to avoid looking at other's code in full, if possible, instead wanting to improve my own. The code I have written runs successfully for the example of and the range 1 to 10, and should be directly modifiable to work with the question.
However, upon running it, I do not get an answer. Presumably, it is a very high number, and the code is not fast enough. Printing the current number being checked seems to support this, reaching several million without getting an answer. I have already made a couple changes which I think should help the speed. For one, for a number to be divisible by all numbers 1 to 20, it must be even, as only even numbers are divisible by 2.
Hence, I can increment by 2 instead of 1. Also, although I didn't think of it myself, I found someone point out that a number divisible by 11 to 20 is divisible by 1 to Haven't checked that one, but it seems reasonable. The code still, however is not fast enough.
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What optimisations, either programmatic, or mathematics, can I make to make this code run faster? Taking the advice of Michael Mior and poke, I wrote a solution. I tried to use a few tricks to make it fast. Since we need a relatively short list of numbers tested, then we can pre-build the list of numbers rather than repeatedly calling xrange or range.
Also, while it would work to just put the numbers [1, 2, 3, If we leave the 20 in, there is no need to leave the 2 in. Any integer evenly divisible by 20 is evenly divisible by 2 but the reverse might not be true.
So we leave the 20 and take out the 2, the 4, and the 5. Leave the 19, as it's prime. Leave the 18, but now we can take out the 3 and the 6.Here are the problems and my commented code for each one in case it interests anybody. If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is Find the sum of all the multiples of 3 or 5 below Each new term in the Fibonacci sequence is generated by adding the previous two terms.
By starting with 1 and 2, the first 10 terms will be: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms. The prime factors of are 5, 7, 13 and What is the largest prime factor of the number ? A palindromic number reads the same both ways. Find the largest palindrome made from the product of two 3-digit numbers. Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.
By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is What is the 10,st prime number? Find the thirteen adjacent digits in the digit number that have the greatest product.
What is the value of this product? Find the sum of all the primes below two million. I am a student and I am having trouble understanding what happens in your solution for problem 5.
And when I plugged in that same code, it printed all whole numbers from 1. Can you explain why that happens? Hope that helps. You are commenting using your WordPress.
Project Euler – Problem 14 (Python Solution)
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Notify me of new posts via email. Problem 1 — Multiples of 3 and 5 If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9.
By starting with 1 and 2, the first 10 terms will be: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms. Took 1. What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20? If it got to 20 it stopped and printed Took about to run.
Second time I wrote it, I started from 1 going up and divided by all the numbers from 20 down to 1 and counted number of multiples and if I got to 20 it stopped and printed Took about seconds to run. This speeds it up because a number is less likely to be divisible by 20 or 19 than by 1 or 2 so by starting at the top you rule out that number quicker and move on.
Code Review Stack Exchange is a question and answer site for peer programmer code reviews. It only takes a minute to sign up. What do you think about my code? I am a beginner and just started doing some algorithmic exercises to sharpen up my python skills.
I'm sure there are better ways to solve those problems but would like to see what you think about my quite simple idea. In comparison, your code has a list of possibly large size.
For bigger n:s the execution time is pretty much unaffected. Regarding the second question, the simple for loop is the best thing and quite fast than the naive recursive algorithm. However the Wikipedia page discusses some great formulas which can speed-up your search Fibonacci Numbers Wiki.
One option is to use Euler himself to solve Euler Project 1. Namely triangular numbers Elements of Algebra, Sign up to join this community. The best answers are voted up and rise to the top. Project Euler problems 1 and 2 in python Ask Question. Asked 3 years, 10 months ago. Active 3 months ago.
Project Euler Problems 1-10 in Python
Viewed 6k times. Here is code for problem 1: """If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is Find the sum of all the multiples of 3 or 5 below By starting with 1 and 2, the first 10 terms will be: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.
Active Oldest Votes. Graipher Graipher I didnt know anything about generators until now but after reading some posts on SO about them I think I kinda get what they are about, definitely will be looking more into them - thank you again for feedback!
If after some time usually at least a day is recommended to give people in other timezones also the chance to answeryou find that you like one answer best, you can accept it using the green checkmark to the left of the post. Many eulers is like that. Instead of creating the numbers, add the all at the same time :. Simon Simon 5 5 silver badges 15 15 bronze badges. Vishesh Mangla Vishesh Mangla 11 11 bronze badges. Calculate number of occurances of num within range.
Then calculatate triangular number of occurances and multiply by num. Aivar Paalberg Aivar Paalberg 2 2 bronze badges.
Sign up or log in Sign up using Google.Here I make my solutions publicly available for other enthusiasts to learn from and to critique. Each problem that I solved always includes a Java program. Almost all my solved problems also include a Python program except for a few.
Many problems additionally have a Mathematica and Haskell program. Among the web, this is perhaps the largest collection of Project Euler solutions in the Java programming language.
Every Java solution depends on these shared classes: EulerSolution. Many Python solutions depend on my shared math library module: eulerlib. Many Haskell solutions depend on my shared math library module: EulerLib. Some solution code contains a detailed mathematical proof of correctness. For every problem that I solved, I have a Java solution for it and possibly code in other languages as well.
I like using Java because it is fast, safe, and expressive. Custom data structures like graphs are difficult to express cleanly in Mathematica. Custom algorithms like the sieve of Eratosthenes, especially ones most naturally expressed in terms of imperative state updates, are difficult to implement correctly or efficiently in Haskell.
Non-strict evaluation in Haskell makes it easy to accidentally leak large amounts of memory in unexpected places. This is because it has many useful built-in mathematical functions like prime testing and objects like fractions that would require manual effort to implement in Java.
Also, my Java solutions tend to be long due to types on every variable, lots of custom code, and low-level loops instead of higher-order functions. Note that for problems involving non-whole numbers, I try to use exact integer arithmetic or fractions as much as possible, which ensures that the solution is provably correct. As a result I strongly avoid any floating-point arithmetic at all, unless there is no other reasonable way that I know of to solve the problem. Also I study the numerical bounds carefully to avoid integer overflow, and use the most reasonably narrow type for speed choosing between intlongor BigInteger.
To run a Java solution, compile the Java file e. Then run with a command like java pand the answer will be printed to standard output.
Project Euler - Problem 1
Almost all solutions are made available in Python, which is an imperative object-oriented language with many conceptual similarities to Java. My code requires Python 3 but old versions can be found that support both 2 and 3. The Python solutions were initially based on the Java solutions, often starting with a direct literal port of the Java code into Python.
Mathematica provides easy access to prime numbers, big integers, high-precision floats, fractions, continued fractions, and more. My code tends to be quite short: one-liners are very common, and typically the solution is less than 5 lines of code.
Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. I'm not looking to use a different method, I'm wondering why this code isn't working as is, from the debugging that I did it looks as everything that I am expecting to see is there. Your solution includes numbers that are multiples of both 3 and 5, twice. You are adding 15, 30, 45, etc. Your solution can be fixed by testing if b is already present in numArray :.
Together with the sum built-in function and a generator expression, that becomes:. Both approaches still loop and thus will take more time as the numbers grow. There is however a mathematical solution that takes constant time. I doubt anyone who doesn't take pleasure in Project Euler can stick with it. If it's not clear what's fun about it, here's a hint. When your earlier attempt at a problem fails, take it as a puzzle to work out what you may need to do differently.
If you still get a little impatient, you can generally search out a principle behind the problem. For example, by the time you get to one particular problem, you'll probably know that problems with integer parameters and integer answers are called "Diophantine". You'll know that problems with squares of variables are called "quadratic".
By searching those words you'll find that the problem is a "Pell equation". In "normal" python there are no arrays. They are called lists. To use arrays one has to install librarys like numpy. A list adjusts its lenght to the need so you dont have to know how big it will be when you define it. This has been a while back but hopefully by now you can reformulate your code into a convenient 1-liner such as:. Learn more. Project Euler 1 in Python Ask Question. Asked 6 years, 8 months ago.
Active 2 years, 3 months ago. Viewed 2k times.
Problem 1 Project Euler must return is returning I'm not looking to use a different method, I'm wondering why this code isn't working as is, from the debugging that I did it looks as everything that I am expecting to see is there.
Active Oldest Votes. Thank you for this! I feel silly, I was testing it with the example of 10 and getting the result that they got and got very confused when it got to higher numbers.
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