Solution:

The question can practically disarm you. But we promise by the time you read the solution, you will find that it was way easy than you were thinking.

Let us name the boxes with a number - Box1, Box 2..., Box10. Now you must be familiar with the weights of the balls precisely. By saying that we are not implying that you will have to take one ball from every box and judge the weight of every ball.

What you have to do is pick one ball from Box 1, 2 balls from Box 2, 3 balls from Box 3, ...., 10 balls from Box 10. Thus after taking balls from every box in such a manner, you will finally have 55 balls in total. There comes the time to use the weighing machine.

If all the balls were weighing adequate, the combined weight should be 55 x 10 grams = 550 grams. But since one of the box has defected balls, the weight will be less than that. Here is the tricky part and explains why we took different number of balls from each box.

If the total weight is less than 1 gram, then the defective box is the Box 1 since we took 1 ball from that box. If the Box 2 is defective, the total weight will be less than 2 grams. In similar fashion you can identify the defective box by analyzing the total weight that is calculated by the weighing machine.

Solution:

Were you thinking of using the fastest person who takes one minute to travel to and fro till everyone has reached the other end? Yes that can be done but it will take 10 + 1 + 7 + 1 + 2 = 21 minutes in total.

If we pay attention towards finding a way to merge the one taking 7 minutes and the one taking 10 minutes, we will have to acknowledge the fact that one of them will also have to return back which will take much more time than 21 minutes.

What if we use the person taking 2 minutes to escort the one taking 1 minutes across? Let's come to the conclusion:

1 and 2 cross the bridge
2 returns with the torch
7 and 10 cross the bridge
1 comes back
1 and 2 cross the bridge

In such a manner, the total time taken will be
2 + 2 + 10 + 1 + 2 = 17 minutes.

Solution:

This puzzle is also known as Monty Hall Puzzle. If Michael choose the box on the table, the probability of finding the pearl inside will be 2/3.

Let us consider the same thing with more boxes. If Sarah placed 100 boxes on the table with one box that contains the pearl and Michael is allowed to pick random, 99 boxes will remain on the table. Now at least 98 boxes in the table will be empty. Sarah will remove these 98 boxes and now Michael will have to choose from the box on table or the box in his hand. If he chooses the box below, the probability of finding the pearl in the box will be 99/100.

Now you must be convinced why he should pick the box on the table.

Solution:

Fill the 3 quart bucket (5 quart bucket has 0 quarts and 3 quart bucket has 3 quarts)

Pour it entirely in the 5 quart bucket (5 quart bucket has 3 quarts and 3 quart bucket has 0 quarts)

Fill the 3 quart bucket (5 quart bucket has 3 quarts and 3 quart bucket has 3 quarts)

Pour the contents to fill the 5 quart bucket (5 quart bucket has 5 quarts and 3 quart bucket has 1 quart)

Empty down the 5 quart bucket (5 quart bucket has 0 quarts and 3 quart bucket has 1 quart)

Pour the water from 1 quart bucket to 5 quart bucket (5 quart bucket has 1 quart and 3 quart bucket has 0 quart)

Fill the 3 quart bucket (5 quart bucket has 1 quart and 3 quart bucket has 3 quarts)

Pour the contents from 3 quart bucket to 5 quart bucket (5 quart bucket has 4 quarts and 3 quart bucket has 0 quarts)

We are done and it took us eight steps to measure exactly 4 quarts of water.

Solution:

Alpha has Yellow
Beta has Yellow
Gamma has Green
Delta has Red

Explanation:
To solve this problem one must keep in mind that
a) Two are lying and two are telling truth
b) Neither two sides of a card are of same color nor two cards are similar
On trying all the possible combinations while keeping these two situations in mind.