看差值选两张梅花牌：最大保证收益

Card Diff

专题: Finance / 金融
难度: L6
来源: QuantQuestion

题目详情

你手里只有一副牌的梅花（共 13 张）。规定 A=1，J/Q/K=0。

你最终必须选两张牌，收益为两张牌数值的乘积。

你可以花 1 元来揭示任意两张牌数值的差（可以重复多次、对任意对）。

在最优策略下，你能保证的最大净收益是多少？

You have all the clubs from a standard deck of cards, which consists of $13$ cards. You can choose $2$ cards from the deck, and your payout is the product of their values. Note that all face cards are assigned a value of $0$ , while the Ace has a value of $1$ . You can pay $1 to reveal the difference in value between any two selected cards. This "difference" option can be exercised repeatedly on any pair of cards. Ultimately, you must choose two cards for which you will calculate the product to determine your payout. With a rational strategy, what is the maximum guaranteed profit you can achieve?

解析

可以在最多花费 11 次“看差值”的成本下，确定 9 与 10 的位置，从而确保最后选到 9 与 10。

最终保证净收益为

9\cdot 10-11=79.

Original Explanation

The claim is that you can always identify the positions of the $9$ and $10$ within $11$ card draws, yielding a payout of $9 \cdot 10 - 11 = 79$ . When you select a card, you pay $1$ to reveal the difference between it and each of the remaining $12$ cards. There are two cases: the first card is either $0$ or not. If the first card is $0$ , you have $12$ cards to compare against and within $11$ draws, you will either find two $0$ differences (identifying the card as $0$ ) and a $9$ difference, or a $10$ difference (or both). Thus, you would only need to spend $11$ , as the last unchecked card must be the one corresponding to the missing difference. If the first card has a value other than $0$ , there are still $3$ cards with a value of $0$ . You can identify your card when you reveal $2$ face cards, both valued at $0$ , which yield the same difference. In this case, you also need to pay at most $11$ to determine the locations of the $9$ and $10$ . As we can always find the $9$ and $10$ cards with a cost of $11$ , our final answer becomes $(\$9 \times \$10) - \$11 = \$79$ .