# Questions tagged [project-euler]

Project Euler is a series of challenging mathematical/computer programming problems. Please see the site and rules before posting.

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### Mathematical Application of Project Euler 215 - Crack-free Walls

I am taking a Math course as part of a Data Science curriculum. For the final project we have to solve a math problem in Python and write it up. I chose Project Euler 215 Crack-free Walls, my course ...
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### Issues generating all primitive integer triangles with 60 degree angle (Eisenstein Triples)

I am trying to write a program that generates all primitive Eisenstein Triples, i.e. triangles with integer sides that have a 60 degree angle. These triplets satisfy the equation $a^2-ab+b^2=c^2$. ...
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### Powers of $2$ starting with $123$…Does a pattern exist?

I'm currently working on Project Euler problem #686 "Powers of Two". The first power of $2$ which starts with $123$... is $2^{90}$. I noticed that the next powers of $2$ that start with $123$...
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### Transforming two 3D vectors into each other gives different Euler angles

Lets take two vectors A and B represented in x,y,z(metres) and Euler angles XYZr(deg) as ...
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### Finding values for which sequence $x_{n+1}=x_n-1/x_n$ is periodic

I'm looking at the sequence $x_{n+1}=x_n-1/x_n$ and want to find starting values $x_0$ for which the sequence is periodic. I want to calculate all values for which the sequence is periodic with a ...
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### Factors of $2n^2 \leq n$?

How many factors of $2n^2$ are less than or equal to $n$? I know that the number of factors of $n^2$ less than $n$ is half the number of factors of $n^2$ (each factor $< n$ corresponds with one ...
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### Maximum-sum path down a triangle of numbers

We have a triangle of $n$ positive and integer number. we start from top or (head) of this triangle and in each step we are going to adjacent number in next row. goal is finding path of maximum ...
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### Project Euler #731_part2 [closed]

In this link : Project Euler problem #731 , I got a nice solution for my question but i'm missing a crucial part . I want to get an explanation of this part of the answer : My question : Why did we ...
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### Is there a way to find an upper bound for $n^2+an+b$?

I was solving the Project Euler: Problem 27. Considering quadratics of the form $n^2 + an + b$, where $|a| \lt 1000$ and $|b| \le 1000$ Find the product of the coefficients, $a$ and $b$, for the ...
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### Project Euler #329 (Prime Frog) - Stochastic independence

Susan has a prime frog.Her frog is jumping around over 500 squares numbered 1 to 500. He can only jump one square to the left or to the right, with equal probability, and he cannot jump outside the ...
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### Why is the one quadratic polynomial a perfect square more often than the other?

I was solving problem 137 of Project Euler, which led me to find $n$ such that $5n^2+2n+1$ is a perfect square. But such numbers are very rare (the 13th is around 3 billions) so after decomposing into ...
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### Is there a way to find a bound on the largest prime factor a given number n?

I worked on Project Euler problem 3 (find largest prime factor of 600851475143) a while back and have tweaked with the code a few times to reuse for other problems, but I eventually found that there ...
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### Spider Fly Problem

I am trying to solve Project Euler problem 86. I know I have to generate Pythagorean triplets and so on. But I have a problem with choosing valid cuboids. For example if the dimensions of the cuboid ...
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### How to solve for var r in closed-form arithmetic-geometric function? (Euler problem 235)

Disclaimer: Math noob here. I have this arithmetic-geometric sequence $u(k)=(a-dk)r^{k-1}$ and summation $s(n)=\sum_{k=1}^n u(k)$ Using Wikipedia (sources 1, 2, and 3), I have solved for the closed-...
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### How does a < b < c and a + b + c = s imply a < s/3 and b < s/2?

I've been trying to understand the overview given for problem 9 on Project Euler and it mentions that the upper bound for iterating through possible values of a is (s - 3)/3 and for b it is (s - a)/2. ...
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### Combinatorics question based on ProjectEuler 606

Motivation The following text is from Problem 606 from Project Euler : A gozinta chain for $n$ is a sequence $\{1,a,b,...,n\}$ where each element properly divides the next. For example, there ...
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### Amount of numbers not divisible by 7 in Pascals Triangle without iteration

For project Euler 148 problem, I want to get the amount of numbers in Pascals Triangle that are not divisible by 7 in row 0 to n where n is $10^9$. Find the number of entries which are not divisible ...
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### project euler #577 confusion

Problem Link. An equilateral triangle with integer side length $n≥3$ is divided into $n^2$ equilateral triangles with side length 1 as shown in the diagram in the above link. The vertices of these ...
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### Modeling path of a rolling ellipse

I'm trying to solve Project Euler problem 525. My approach is to find a parametric equation that can model the path of the center point as it rolls, then take the arc length of that function for one ...
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### How to evaluate closed form of these series of sum? $\sum_{k=1}^n k*10^{k-1}$

$$\sum_{k=1}^n k*10^{k-1}$$ I came across this summation of series while I was trying to solve Project Euler Problem 40. The problem can be solved without using this method; however, I want to know ...