---
id: 5900f4b71000cf542c50ffca
title: 'Problem 331: Cross flips'
challengeType: 5
forumTopicId: 301989
dashedName: problem-331-cross-flips
---

# --description--

N×N disks are placed on a square game board. Each disk has a black side and white side.

At each turn, you may choose a disk and flip all the disks in the same row and the same column as this disk: thus $2 × N - 1$ disks are flipped. The game ends when all disks show their white side. The following example shows a game on a 5×5 board.

<img class="img-responsive center-block" alt="animation showing game on 5x5 board" src="https://cdn.freecodecamp.org/curriculum/project-euler/cross-flips.gif" style="background-color: white; padding: 10px;">

It can be proven that 3 is the minimal number of turns to finish this game.

The bottom left disk on the $N×N$ board has coordinates (0, 0); the bottom right disk has coordinates ($N - 1$,$0$) and the top left disk has coordinates ($0$,$N - 1$).

Let $C_N$ be the following configuration of a board with $N × N$ disks: A disk at ($x$, $y$) satisfying $N - 1 \le \sqrt{x^2 + y^2} \lt N$, shows its black side; otherwise, it shows its white side. $C_5$ is shown above.

Let $T(N)$ be the minimal number of turns to finish a game starting from configuration $C_N$ or 0 if configuration $C_N$ is unsolvable. We have shown that $T(5) = 3$. You are also given that $T(10) = 29$ and $T(1\\,000) = 395\\,253$.

Find $\displaystyle \sum_{i = 3}^{31} T(2^i - i)$.

# --hints--

`crossFlips()` should return `467178235146843500`.

```js
assert.strictEqual(crossFlips(), 467178235146843500);
```

# --seed--

## --seed-contents--

```js
function crossFlips() {

  return true;
}

crossFlips();
```

# --solutions--

```js
// solution required
```