freeCodeCamp/curriculum/challenges/english/10-coding-interview-prep/project-euler/problem-295-lenticular-holes.md

id, title, challengeType, forumTopicId, dashedName

id	title	challengeType	forumTopicId	dashedName
5900f4931000cf542c50ffa6	Problem 295: Lenticular holes	5	301947	problem-295-lenticular-holes

--description--

We call the convex area enclosed by two circles a lenticular hole if:

• The centres of both circles are on lattice points.
• The two circles intersect at two distinct lattice points.
• The interior of the convex area enclosed by both circles does not contain any lattice points.

Consider the circles:

$$\begin{align} & C_0: x^2 + y^2 = 25 \\ & C_1: {(x + 4)}^2 + {(y - 4)}^2 = 1 \\ & C_2: {(x - 12)}^2 + {(y - 4)}^2 = 65 \end{align}$$

The circles C_0, C_1 and C_2 are drawn in the picture below.

C_0 and C_1 form a lenticular hole, as well as C_0 and C_2.

We call an ordered pair of positive real numbers (r_1, r_2) a lenticular pair if there exist two circles with radii r_1 and r_2 that form a lenticular hole. We can verify that (1, 5) and (5, \sqrt{65}) are the lenticular pairs of the example above.

Let L(N) be the number of distinct lenticular pairs (r_1, r_2) for which 0 < r_1 ≤ r_2 ≤ N. We can verify that L(10) = 30 and L(100) = 3442.

Find L(100\\,000).

--hints--

lenticularHoles() should return 4884650818.

assert.strictEqual(lenticularHoles(), 4884650818);

--seed--

--seed-contents--

function lenticularHoles() {

  return true;
}

lenticularHoles();

--solutions--

// solution required