**How much are the birds?**

Since the price of 1 large bird is equal to the price of 2 small ones, then 5 large birds will cost as much as 10 small ones. This means that 5 large birds plus 3 small ones will cost as much as 13 small ones. On the other hand, the price of 3 large birds and 5 small ones is equal to the cost of 11 small birds. Thus, the difference between the price of 5 large birds and 3 small ones turns out to be equal to the difference between the price of 13 and 11 small birds; that is, it’s equal to the price of 2 small birds. Since 2 small birds cost $20, then the price of 1 small bird equals $10, and a large one costs $20.

**Profit**

Imagine that the trader began his working day with $100. He first bought the shares for $7, leaving him with $93. Then he sold them for $8, following which he had $101. Then he bought back the same shares for $9, so he had $92. Finally, he sold the shares for $10. Ultimately, his account had $102 in it.

**A bear’s fur**

The fur must be white because it belongs to a polar bear living in the Arctic, close to the North Pole. Whatever direction you step in away from the North Pole, you will always move south. So if the bear is at the North Pole, and the hunter is 100 m to the south of it, then by moving 100 m to the east and turning to face the north, the hunter ends up facing the North Pole. There are also other ways to solve this task. For example, the hunter might be located at the 100 m parallel, and the bear is 100 m to the north of him. By moving 100 m to the east, the hunter completes a full circle around the pole and returns to his starting position. This is the second way to solve the task. But the hunter might be still closer to the pole, at the 50 m parallel. By moving 100 m, he circles the pole twice and finds himself back in his starting position. But that’s not all. The hunter might be at 1/3 of the 100 m parallel. By walking 3 times around the pole, he ends up back in his starting position.

**Answer to an Impossible Escape**

They can actually escape! Here’s how to think about it from each person’s perspective.

What drives the puzzle is that each time a person passes, that signals information to the other player which transforms some information into common knowledge.

Day 1

If Alice saw 19 or 20 trees, then Alice could conclude there are 20 trees. But Alice only sees 12, so Alice passes. This signals to Bob that Alice sees at most 18 trees.

If Bob saw 0 or 1 trees, combined with the fact Alice sees at most 18 trees, he could conclude there would have to be 18 trees. But Bob sees 8 trees. So Bob has to pass. This signals to Alice that Bob sees at least 2 trees.

Day 2

If Alice saw 17 or 18 trees, then Alice could conclude there are 20 trees because Bob must see at least 2 trees. But Alice only sees 12, so Alice passes. This signals to Bob that Alice sees at most 16 trees.

Now if Bob saw 2 or 3 trees, combined with the fact Alice sees at most 16 trees, he could conclude there would have to be 18 trees. But Bob sees 8 trees. So Bob has to pass. This signals to Alice that Bob sees at least 4 trees.

Day 3

If Alice saw 15 or 16 trees, then Alice could conclude there are 20 trees because Bob must see at least 4 trees. But Alice only sees 12, so Alice passes. This signals to Bob that Alice sees at most 14 trees.

Now if Bob saw 4 or 5 trees, combined with the fact Alice sees at most 14 trees, he could conclude there would have to be 18 trees. But Bob sees 8 trees. So Bob has to pass again. This signals to Alice that Bob sees at least 6 trees.

Day 4

If Alice saw 13 or 14 trees, then Alice could conclude there are 20 trees because Bob must see at least 6 trees. But Alice only sees 12, so Alice passes. This signals to Bob that Alice sees at most 12 trees.

Now if Bob saw 6 or 7 trees, combined with the fact Alice sees at most 12 trees, he could conclude there would have to be 18 trees. But Bob sees 8 trees. So Bob has to pass again. This signals to Alice that Bob sees at least 8 trees.

Day 5

Since Bob sees at least 8 trees, and Alice sees 12 trees, Alice knows there are 20 trees. She guesses 20 and they are freed!