Reasoning-Let your brain fly

I,
Mr. P and Mr. Q both have sufficient reasoning capabilities. On this day, they were waiting for a reasoning interview.
They know that there are 16 cards in the drawer of the table:
Peach A, Q, 4
Taotao J, 8, 4, 2, 7, 3
Caohua K, Q, 5, 4, 6
Square A, 5

Professor John picked a card from the 16 cards and told Mr. P the points of the card.
Tell Mr. Q.
Professor John asked Mr. P and Mr. Q: Can you tell from the known points or colors what the card is?
Mr. P: "I don't know this card. "
Mr. Q: "I know you don't know this card. "
Mr. P: "Now I know this card. "
Mr. Q: "I know. "
Excuse me: What is this card?

II,
Five pirates snatched 100 gems, each of which was of the same size and value.
They decided:
1. draw lots to determine your own number (1, 2, 3, 4, 5)
2. First, the allocation scheme is proposed by No. 1, and then five people vote. If only half and more than half of the people agree to the allocation scheme, otherwise, they will be thrown into the sea to feed sharks.
3. If the first day after the death of the first day, the second day will propose the allocation scheme, and then four people will vote. If and only half and more than half of the people agree, distribute as per his proposal, otherwise they will be thrown into the sea to feed sharks.
4. analogy ......
Condition:
Every pirate is a very intelligent person who can make decisions by making rational judgments on gains and losses.
Problem:
What is the allocation scheme proposed by the first pirate to maximize its own benefits?

III,
Each of the 50 people in the village has a dog. Now, they know that the dog is ill. Each person can observe the dog of others, but cannot observe his or her own, and cannot communicate with each other, it was inferred that he was a sick dog and then shot it down that day. On the first day, there was no movement on the second day. On the third day, there was a gunshots and asked how many sick dogs were there.

IV,
The professor took three white hats and two black hats and told the three students to wear a hat for each of them. Each person could see the other two and could not see their own, guess the color of your hat. After wearing it, the three looked at each other for a while, and then said in a different voice that they were wearing white hats. The professor smiled with satisfaction. Why?

V,
There are a bunch of hats, which contain seven known colors. Now seven people are wearing one, and they can be the same color. Everyone can see the colors of others' hats, but not those of their own. They are not allowed to communicate, guess their own colors, and do not let others know. Before wearing a hat, you can discuss whether there is a strategy to make sure at least one of them will guess right?

VI,
A pair of playing cards, remove the cat, the maximum is A, the minimum is 2, no color difference. Play with four people. Each person takes a card and shows it to others. But I don't know what this card is. When a person has more cards than all others, he wins because there is no color difference, so there is a possibility. Deduce in sequence, and draw the highest-level conclusion based on the inference (possible conclusions are: "I don't know", "I win", and "I lose ", "I did not win", "I did not lose", "I lost "). Now there is a situation where four people in the first round say they don't know. The second round is still like this. The first three people in the third round still say they don't know what they will say, what cards are in his hand, and what cards are in others' hands.

VII,
The teacher picked two integers from 2 to 40, told them the sum, and told the product to B, and told everyone that the product is not greater than 200. A told B: You certainly don't know the number in my hand. B: Now I know. A said: I know your number too. Q: What are the two numbers?

8,
The teacher wrote the three numbers on three pieces of paper and pasted them on the heads of the three students respectively, telling them that the three numbers are all positive integers, one of which is the sum of the other two numbers. Each person can see the number of the other two and cannot see their own. Now I guess the number on my head. The first student said, "I don't know." The second and third students also said they don't know. Then asked the first student, who said no, but the second student still said no. The third student said, "I know, It's 144 ". What are the other two numbers?