Here is the English translation of the given content: "169 Classic Games to Open the Wisdom of Teenagers"

Author: Gao Yusheng, Xu Baoping / Country: Mainland China
Publisher:
Publish Date: 2006-08-01
Features: Wisdom is power, wisdom is wealth, wisdom is the foundation of life success. Games are learning, games are fun, games are the driving force for intellectual development. Wisdom is power, wisdom is wealth, wisdom is the foundation of life success. Games are learning, games are fun, games are the driving force for intellectual development.
96 Rabbits and Wolves
When the rabbit is at the center of the lake and the wolf is at some point on the shore, if the rabbit directly rows to the shore, it is best to row to the symmetric point of the wolf on the shore, because this point is the farthest from the wolf. At this time, the distance the rabbit rows is R, while the wolf needs to run half a circle around the lake to intercept the rabbit on the shore. The distance of half a circle is 3.14R. Since the wolf's speed is four times the rabbit's rowing speed, the wolf can reach the rabbit's landing point before the rabbit reaches the shore and wait there to catch the rabbit. It seems that using this method, the rabbit cannot escape. In fact, the rabbit has a clever strategy to reach the shore first. It can first row the boat to some point on a circle with the lake center as the center and a radius of 0.24R. Then, it rows counterclockwise (or clockwise) along this smaller circle. Since the radius of the smaller circle is less than 0.25R, the circumference of this smaller circle is also less than 0.25 of the lake shore's circumference. The rabbit's angular velocity when rowing around the small circle is slightly greater than the wolf's angular velocity when running along the lake shore. Therefore, even if the wolf starts closest to the rabbit on the large shore, the rabbit can gradually pull ahead by rowing around the small circle, and the wolf will slowly fall behind. Eventually, there will be a moment when the wolf is on the far side of the shore from the rabbit. At this point, the rabbit and the wolf are on opposite sides of a diameter, and the rabbit's closest distance to the shore is 0.76R, while the wolf needs to run half the lake's circumference (3.14R) to reach that point. Since 3.14R > 4 × 0.76R, the wolf cannot reach that point before the rabbit rows to shore. The rabbit can then safely escape by stepping onto the shore.
97 Pirate Gold Division
The key to analyzing all such strategy games lies in working backward from the end. When the game ends, it is easy to determine which decisions are favorable and which are unfavorable. Once this is clear, you can apply it to the second-to-last decision, and so on. If you analyze the game from the beginning, you won’t get very far. The reason is that all strategic decisions involve determining: "If I do this, what will the next person do?" Therefore, the decisions made by the pirates after you are important to you, while the decisions made by the pirates before you are not, because you have no control over them anyway. Remembering this, you can see that our starting point should be when the game has only two pirates left (i.e., Pirate 1 and Pirate 2). At this point, the most formidable pirate is Pirate 2, and his best distribution plan is obvious: all 100 gold coins go to him, and Pirate 1 gets nothing. Since he will certainly vote in favor of this plan, it secures 50% of the votes, so the plan passes.
Now, let’s add Pirate 3. Pirate 1 knows that if Pirate 3’s plan is rejected, the game will then have only two pirates left, and Pirate 1 will certainly get nothing. Additionally, Pirate 3 knows that Pirate 1 is aware of this situation. Therefore, as long as Pirate 3 offers a small enough bribe to Pirate 1 to ensure he doesn’t walk away empty-handed, Pirate 1 will vote in favor of any plan Pirate 3 proposes. Thus, Pirate 3 needs to bribe Pirate 1 with the smallest possible amount of gold, which leads to the following distribution plan: Pirate 3 keeps 99 gold coins, Pirate 2 gets nothing, and Pirate 1 gets 1 gold coin.
Pirate 4’s strategy is similar. He needs 50% of the votes, so like Pirate 3, he must find another pirate to support him. He can bribe the lowest amount of 1 gold coin to Pirate 2. Since if Pirate 4’s plan is rejected and Pirate 3’s passes, Pirate 2 will get nothing. Therefore, Pirate 4’s distribution plan should be: 99 gold coins for himself, Pirate 3 gets nothing, Pirate 2 gets 1 gold coin, and Pirate 1 gets nothing.
Pirate 5’s strategy is slightly different. He needs to bribe two other pirates, so he must spend at least 2 gold coins to ensure his plan passes. His distribution plan should be: 98 gold coins for himself, 1 gold coin for Pirate 3, and 1 gold coin for Pirate 1.
This analysis process can continue along these lines. Each distribution plan is uniquely determined, maximizing the proposer’s gold while ensuring the plan passes. Following this pattern, Pirate 10’s proposed plan would be: 96 gold coins for himself, 1 gold coin each for the even-numbered pirates, and nothing for the odd-numbered pirates. This solves the distribution problem for the 10 pirates.
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