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leetcode-cn/problem (English)/猴子碰撞的方法数(English) [count-collisions-of-monkeys-on-a-polygon].html
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<p>There is a regular convex polygon with <code>n</code> vertices. The vertices are labeled from <code>0</code> to <code>n - 1</code> in a clockwise direction, and each vertex has <strong>exactly one monkey</strong>. The following figure shows a convex polygon of <code>6</code> vertices.</p>
<img alt="" src="https://assets.leetcode.com/uploads/2023/01/22/hexagon.jpg" style="width: 300px; height: 293px;" />
<p>Each monkey moves simultaneously to a neighboring vertex. A neighboring vertex for a vertex <code>i</code> can be:</p>
<ul>
<li>the vertex <code>(i + 1) % n</code> in the clockwise direction, or</li>
<li>the vertex <code>(i - 1 + n) % n</code> in the counter-clockwise direction.</li>
</ul>
<p>A <strong>collision</strong> happens if at least two monkeys reside on the same vertex after the movement or intersect&nbsp;on an edge.</p>
<p>Return <em>the number of ways the monkeys can move so that at least <strong>one collision</strong></em> <em> happens</em>. Since the answer may be very large, return it modulo <code>10<sup>9 </sup>+ 7</code>.</p>
<p><strong>Note</strong> that each monkey can only move once.</p>
<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<pre>
<strong>Input:</strong> n = 3
<strong>Output:</strong> 6
<strong>Explanation:</strong> There are 8 total possible movements.
Two ways such that they collide at some point are:
- Monkey 1 moves in a clockwise direction; monkey 2 moves in an anticlockwise direction; monkey 3 moves in a clockwise direction. Monkeys 1 and 2 collide.
- Monkey 1 moves in an anticlockwise direction; monkey 2 moves in an anticlockwise direction; monkey 3 moves in a clockwise direction. Monkeys 1 and 3 collide.
It can be shown 6 total movements result in a collision.
</pre>
<p><strong class="example">Example 2:</strong></p>
<pre>
<strong>Input:</strong> n = 4
<strong>Output:</strong> 14
<strong>Explanation:</strong> It can be shown that there are 14 ways for the monkeys to collide.
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>
<ul>
<li><code>3 &lt;= n &lt;= 10<sup>9</sup></code></li>
</ul>
<p>There is a regular convex polygon with <code>n</code> vertices. The vertices are labeled from <code>0</code> to <code>n - 1</code> in a clockwise direction, and each vertex has <strong>exactly one monkey</strong>. The following figure shows a convex polygon of <code>6</code> vertices.</p>
<img alt="" src="https://assets.leetcode.com/uploads/2023/01/22/hexagon.jpg" style="width: 300px; height: 293px;" />
<p>Simultaneously, each monkey moves to a neighboring vertex. A <strong>collision</strong> happens if at least two monkeys reside on the same vertex after the movement or intersect on an edge.</p>
<p>Return the number of ways the monkeys can move so that at least <strong>one collision</strong> happens. Since the answer may be very large, return it modulo <code>10<sup>9 </sup>+ 7</code>.</p>
<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 3</span></p>
<p><strong>Output:</strong> <span class="example-io">6</span></p>
<p><strong>Explanation:</strong></p>
<p>There are 8 total possible movements.<br />
Two ways such that they collide at some point are:</p>
<ul>
<li>Monkey 1 moves in a clockwise direction; monkey 2 moves in an anticlockwise direction; monkey 3 moves in a clockwise direction. Monkeys 1 and 2 collide.</li>
<li>Monkey 1 moves in an anticlockwise direction; monkey 2 moves in an anticlockwise direction; monkey 3 moves in a clockwise direction. Monkeys 1 and 3 collide.</li>
</ul>
</div>
<p><strong class="example">Example 2:</strong></p>
<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 4</span></p>
<p><strong>Output:</strong> <span class="example-io">14</span></p>
</div>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>
<ul>
<li><code>3 &lt;= n &lt;= 10<sup>9</sup></code></li>
</ul>