freeCodeCamp/curriculum/challenges/english/10-coding-interview-prep/project-euler/problem-450-hypocycloid-and-lattice-points.md

id, title, challengeType, forumTopicId, dashedName

id	title	challengeType	forumTopicId	dashedName
5900f52e1000cf542c510041	Problem 450: Hypocycloid and Lattice points	5	302123	problem-450-hypocycloid-and-lattice-points

--description--

A hypocycloid is the curve drawn by a point on a small circle rolling inside a larger circle. The parametric equations of a hypocycloid centered at the origin, and starting at the right most point is given by:

x(t) = (R - r) \cos(t) + r \cos(\frac{R - r}{r}t) y(t) = (R - r) \sin(t) - r \sin(\frac{R - r}{r} t)

Where R is the radius of the large circle and r the radius of the small circle.

Let C(R, r) be the set of distinct points with integer coordinates on the hypocycloid with radius R and r and for which there is a corresponding value of t such that \sin(t) and \cos(t) are rational numbers.

Let S(R, r) = \sum\_{(x,y) \in C(R, r)} |x| + |y| be the sum of the absolute values of the x and y coordinates of the points in C(R, r).

Let T(N) = \sum_{R = 3}^N \sum_{r=1}^{\left\lfloor \frac{R - 1}{2} \right\rfloor} S(R, r) be the sum of S(R, r) for R and r positive integers, R\leq N and 2r < R.

You are given:

$$\begin{align} C(3, 1) = & \{(3, 0), (-1, 2), (-1,0), (-1,-2)\} \\ C(2500, 1000) = & \{(2500, 0), (772, 2376), (772, -2376), (516, 1792), (516, -1792), (500, 0), (68, 504), \\ &(68, -504),(-1356, 1088), (-1356, -1088), (-1500, 1000), (-1500, -1000)\} \end{align}$$

Note: (-625, 0) is not an element of C(2500, 1000) because \sin(t) is not a rational number for the corresponding values of t.

S(3, 1) = (|3| + |0|) + (|-1| + |2|) + (|-1| + |0|) + (|-1| + |-2|) = 10

T(3) = 10; T(10) = 524; T(100) = 580\\,442; T({10}^3) = 583\\,108\\,600.

Find T({10}^6).

--hints--

hypocycloidAndLatticePoints() should return 583333163984220900.

assert.strictEqual(hypocycloidAndLatticePoints(), 583333163984220900);

--seed--

--seed-contents--

function hypocycloidAndLatticePoints() {

  return true;
}

hypocycloidAndLatticePoints();

--solutions--

// solution required