12. In the Tumba-Yumba tribe with a population of 30 people, a trader arrives. After studying the customs of the tribe, the trader proposes to play a game. For each natural exchange of goods conducted in the market by two tribespeople, the trader gives each participant one gold coin. If at the end of the day, two different tribespeople end up with the same number of coins, all goods and coins go to the trader. A tribesperson trades goods only with another known tribesperson, and the cunning trader knows this. By the end of the day, the trader finds a group of tribespeople with the same number of coins. The tribespeople are shocked, the trader is happy, and the offending tribespeople are expelled from the market. Then the Chief appears and offers the trader a rematch. Let the trader distribute 270 coins among the remaining tribespeople so that no one has the same number of coins (i.e., each tribesperson should have a unique number of coins). If the trader cannot do this, all the coins and goods remain in the tribe.

- Is it true that in the original game, the tribespeople had no chance?
- Is it true that the Chief can return all the goods and gold?
- Is it true that the Chief knew how many tribespeople were expelled from the market?
- What is the maximum number of tribespeople that needed to be expelled for the Chief to win?
- Is it true that if the trader had lost the first game, the tribespeople could have lost the second game?

Explain your answers.
Solution: In any company, there are at least two people who have the same number of acquaintances. Therefore, the natives had no chance. The chief proposed to distribute 270 coins, which means he knew that some natives would be removed. Let's say x people were removed. To distribute a different number of coins to everyone, you need to take $\frac{0+30-x-1}{2}(30-x)$ coins, and the Chief should make sure that 270 coins are not enough. Then, at least 24 natives should remain, so $24 * 23 / 2=276$ coins are needed to avoid repetition. Since $23 * 11=253$ contains an arithmetic error, which does not affect the result. The correct answer to 2, 3, or 4 questions | 5 |
| The correct answer and justification for only one of the questions in the problem | 2 |
| The answer is either incorrect or guessed, with no justification | 0 |