---
id: 5900f5191000cf542c51002b
title: 'Problem 428: Necklace of Circles'
challengeType: 5
forumTopicId: 302098
dashedName: problem-428-necklace-of-circles
---
# --description--
Let `a`, `b` and `c` be positive numbers.
Let W, X, Y, Z be four collinear points where |WX| = `a`, |XY| = `b`, |YZ| = `c` and |WZ| = `a` + `b` + `c`.
Let Cin be the circle having the diameter XY.
Let Cout be the circle having the diameter WZ.
The triplet (`a`, `b`, `c`) is called a *necklace triplet* if you can place `k` ≥ 3 distinct circles C1, C2, ..., Ck such that:
- Ci has no common interior points with any Cj for 1 ≤ i, j ≤ k and i ≠ j,
- Ci is tangent to both Cin and Cout for 1 ≤ i ≤ k,
- Ci is tangent to Ci+1 for 1 ≤ i < k, and
- Ck is tangent to C1.
For example, (5, 5, 5) and (4, 3, 21) are necklace triplets, while it can be shown that (2, 2, 5) is not.
Let T(`n`) be the number of necklace triplets (`a`, `b`, `c`) such that `a`, `b` and `c` are positive integers, and `b` ≤ `n`. For example, T(1) = 9, T(20) = 732 and T(3000) = 438106.
Find T(1 000 000 000).
# --hints--
`necklace(1000000000)` should return 747215561862.
```js
assert.strictEqual(necklace(1000000000), 747215561862);
```
# --seed--
## --seed-contents--
```js
function necklace(n) {
return true;
}
necklace(1000000000)
```
# --solutions--
```js
// solution required
```