黑屋分牌

Dark Room Deck

专题: Discrete Math / 离散数学
难度: L2
来源: Brainstellar

题目详情

黑暗房间里有 52 张牌，恰好 10 张正面朝上，其余朝下。你可以触摸和翻牌，但无法感知哪张正面朝上。

要求把牌分成两堆，使两堆中“正面朝上的张数”相同。

In a dark room, there is a deck of 52 cards, with exactly 10 cards facing up, rest are facing down. You can touch the cards with your hands but cannot sense which cards are facing up/down. You need to split all these cards into two decks with equal number of cards facing up!

Hint

We never said that both the final decks must have 5 cards each facing up.

解析

做法：

随机拿出 10 张作为 A 堆，其余 42 张为 B 堆。
把 A 堆的 10 张 全部翻面。

证明：设原先 A 堆里有 $x$ 张正面，则 B 堆有 $10-x$ 张正面。A 堆翻面后正面数变为 $10-x$ ，与 B 堆相同。

Original Explanation

Create a deck of 10 cards randomly from the original 52-card deck. And then turn over the 10-card deck.

Solution

This puzzle involves a trick, i.e, we can manually flip over any number of cards.

We create a deck of $k=10$ cards randomly selected from the deck.

Suppose the $k$ -card deck included $x$ number of cards that were originally facing up, and $k-x$ cards are facing up in the other deck.

By flipping over this $k$ -card deck, now there are exactly $k-x$ cards facing up in this deck, same as the other deck.

This question can be solved regardless of the original number $k=10$ , as long as we know the count of cards facing up.