Add: Add 2025/8/29

Author: whats2000

3021-Alice and Bob Playing Flower Game/Note.md

@@ -0,0 +1,78 @@
+# 3021. Alice and Bob Playing Flower Game
+
+Alice and Bob are playing a turn-based game on a field, with two lanes of flowers between them. 
+There are `x` flowers in the first lane between Alice and Bob, and `y` flowers in the second lane between them.
+
+The game proceeds as follows:
+
+1. Alice takes the first turn.
+2. In each turn, a player must choose either one of the lane and pick one flower from that side.
+3. At the end of the turn, if there are no flowers left at all, the current player captures their opponent and wins the game.
+
+Given two integers, `n` and `m`, the task is to compute the number of possible pairs `(x, y)` that satisfy the conditions:
+
+- Alice must win the game according to the described rules.
+- The number of flowers `x` in the first lane must be in the range `[1,n]`.
+- The number of flowers `y` in the second lane must be in the range `[1,m]`.
+
+Return the number of possible pairs `(x, y)` that satisfy the conditions mentioned in the statement.
+
+**Constraints:**
+
+- `1 <= n, m <= 10^5`
+
+## 基礎思路
+
+此遊戲每回合只能從其中一條花道摘 **一朵** 花，因此整局的總步數恰為 $x+y$。
+
+- 若 $x+y$ 為 **奇數**，先手 Alice 將走最後一步並獲勝；
+- 若 $x+y$ 為 **偶數**，則換後手 Bob 走最後一步。
+
+因此，我們要計算在 $1 \le x \le n,; 1 \le y \le m$ 下，滿足 **$x+y$ 為奇數** 的配對數目。
+
+令
+
+- $o_n=\lceil n/2\rceil$ 為 $[1,n]$ 中奇數的個數，
+- $e_n=\lfloor n/2\rfloor$ 為 $[1,n]$ 中偶數的個數；
+- $o_m=\lceil m/2\rceil,; e_m=\lfloor m/2\rfloor$ 同理。
+
+使得 $x+y$ 為奇數的配對來自兩種情況：
+
+1. $x$ 奇、$y$ 偶：共有 $o_n \cdot e_m$ 種；
+2. $x$ 偶、$y$ 奇：共有 $e_n \cdot o_m$ 種。
+
+總數為
+
+$$
+o_n e_m + e_n o_m \;=\; \left\lceil \frac{n}{2}\right\rceil \left\lfloor \frac{m}{2}\right\rfloor \;+\; \left\lfloor \frac{n}{2}\right\rfloor \left\lceil \frac{m}{2}\right\rceil
+\;=\; \left\lfloor \frac{nm}{2}\right\rfloor.
+$$
+
+因此答案等於 $\left\lfloor \dfrac{n m}{2} \right\rfloor$，可用一行計算完成。
+
+## 解題步驟
+
+### Step 1：使用推導出的公式計算結果
+
+我們已經知道，Alice 會獲勝當且僅當 $x + y$ 為奇數，而這樣的配對數量等於 $\left\lfloor \dfrac{n \cdot m}{2} \right\rfloor$。
+因此直接套用此公式計算並回傳答案：
+
+```typescript
+function flowerGame(n: number, m: number): number {
+  return Math.floor(n * m / 2);
+}
+```
+
+## 時間複雜度
+
+- 僅常數次算術與一次向下取整，皆為 $O(1)$。
+- 總時間複雜度為 $O(1)$。
+
+> $O(1)$
+
+## 空間複雜度
+
+- 不使用額外資料結構，僅常數暫存。
+- 總空間複雜度為 $O(1)$。
+
+> $O(1)$

3021-Alice and Bob Playing Flower Game/answer.ts

@@ -0,0 +1,3 @@
+function flowerGame(n: number, m: number): number {
+  return Math.floor(n * m / 2);
+}