最后剩下的球
Last Ball
题目详情
袋子里有 20 个蓝球、14 个红球。每次随机取出 2 个球：
- 若两球同色，则放回 1 个蓝球；
- 若两球异色，则放回 1 个红球。
重复直到只剩 1 个球。问最后剩下球的颜色是什么？
如果初始是 20 蓝 13 红呢？
A bag has 20 blue balls and 14 red balls. You randomly take out 2 balls at a time. If both are the same color, you put back 1 blue ball; if they are different colors, you put back 1 red ball. Repeat until only 1 ball remains. What is the color of the last ball? What if initially there are 20 blue and 13 red?
解析
答案：
- 初始 20 蓝 14 红（红球数为偶数）→ 最后剩 蓝球。
- 初始 20 蓝 13 红（红球数为奇数）→ 最后剩 红球。
关键不变量：红球数量的奇偶性在操作下保持不变，因此最终剩球颜色由红球奇偶决定。
Original Explanation
- With 20 blue and 14 red (an even number of reds), the last ball is blue.
- With 20 blue and 13 red (an odd number of reds), the last ball is red.
Each same-color pair removes 2 of that color but adds 1 blue, effectively changing the count of blue by ±1 and red by ±0 or ±2. One can show that the parity of red balls controls the outcome.