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:

90

Split the cube into three by three grids. You will get three layers. Now get to one square from the cell adjacent instead of counting the path. You can now count the paths from the left (x), the right (y) and the level above (z). So the path for any square will be x + y + z.

Solution:

Turn the first switch on and keep it on for a few minutes. Now switch it off and turn on the second switch and enter the room. If you see that the bulb is lit, then you know that the second switch is connected to the bulb. If the bulb is off, then touch it. If the bulb is hot, the first switch is connected with it and if it is cold, then the third switch is connected to it.

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.

Solution:

Keep one coin from the first set and place it on the scale along with the two from the second set etc. If the weight is off by one hundredth of an ounce, then you will know that it is the first set that is faulty and if the weight is off by two hundred of an ounce, then the second set is faulty and so on.