alice-and-bob-playing-flower-game

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Algorithms

Alice and Bob are playing a turn-based game on a circular field surrounded by flowers. The circle represents the field, and there are <code>x</code> flowers in the clockwise direction between Alice and Bob, and <code>y</code> flowers in the anti-clockwise direction between them.

The game proceeds as follows:

<ol>
	<li>Alice takes the first turn.</li>
	<li>In each turn, a player must choose either the clockwise or anti-clockwise direction and pick one flower from that side.</li>
	<li>At the end of the turn, if there are no flowers left at all, the current player captures their opponent and wins the game.</li>
</ol>

Given two integers, <code>n</code> and <code>m</code>, the task is to compute the number of possible pairs <code>(x, y)</code> that satisfy the conditions:

<ul>
	<li>Alice must win the game according to the described rules.</li>
	<li>The number of flowers <code>x</code> in the clockwise direction must be in the range <code>[1,n]</code>.</li>
	<li>The number of flowers <code>y</code> in the anti-clockwise direction must be in the range <code>[1,m]</code>.</li>
</ul>

Return the number of possible pairs <code>(x, y)</code> that satisfy the conditions mentioned in the statement.

&nbsp;
Example 1:

<pre>
Input: n = 3, m = 2
Output: 3
Explanation: The following pairs satisfy conditions described in the statement: (1,2), (3,2), (2,1).
</pre>

Example 2:

<pre>
Input: n = 1, m = 1
Output: 0
Explanation: No pairs satisfy the conditions described in the statement.
</pre>

&nbsp;
Constraints:

<ul>
	<li><code>1 &lt;= n, m &lt;= 105</code></li>
</ul>

flowerGame

Alice and Bob Playing Flower Game

Math

Alice 和 Bob 在一个长满鲜花的环形草地玩一个回合制游戏。环形的草地上有一些鲜花，Alice 到&nbsp;Bob 之间顺时针有 <code>x</code>&nbsp;朵鲜花，逆时针有 <code>y</code>&nbsp;朵鲜花。

游戏过程如下：

<ol>
	<li>Alice 先行动。</li>
	<li>每一次行动中，当前玩家必须选择顺时针或者逆时针，然后在这个方向上摘一朵鲜花。</li>
	<li>一次行动结束后，如果所有鲜花都被摘完了，那么 当前&nbsp;玩家抓住对手并赢得游戏的胜利。</li>
</ol>

给你两个整数&nbsp;<code>n</code>&nbsp;和&nbsp;<code>m</code>&nbsp;，你的任务是求出满足以下条件的所有&nbsp;<code>(x, y)</code>&nbsp;对：

<ul>
	<li>按照上述规则，Alice 必须赢得游戏。</li>
	<li>Alice 顺时针方向上的鲜花数目&nbsp;<code>x</code>&nbsp;必须在区间&nbsp;<code>[1,n]</code>&nbsp;之间。</li>
	<li>Alice 逆时针方向上的鲜花数目 <code>y</code>&nbsp;必须在区间&nbsp;<code>[1,m]</code>&nbsp;之间。</li>
</ul>

请你返回满足题目描述的数对&nbsp;<code>(x, y)</code>&nbsp;的数目。

&nbsp;

示例 1：

<pre>
输入：n = 3, m = 2
输出：3
解释：以下数对满足题目要求：(1,2) ，(3,2) ，(2,1) 。
</pre>

示例 2：

<pre>
输入：n = 1, m = 1
输出：0
解释：没有数对满足题目要求。
</pre>

&nbsp;

提示：

<ul>
	<li><code>1 &lt;= n, m &lt;= 105</code></li>
</ul>