Solution:

10 Bulbs

For this tricky puzzle, you must check how many light bulbs in the row are having an odd number of factors. The first one surely has odd number of factors, the second has even, four has odd. Thus the bulb four and one will remain lit. The bulbs that are going to remain lit are perfect squares as they have an odd number of factors - 1, 4, 9, 16.

Since there are 100 passes, you can go up to 10 times 10 i.e. the square of 10. There are 10 perfect squares available to you - one, two, three, four, five, six, seven, eight, nine and ten. They corresponds to the bulb number 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. All of them will remain lit and thus total ten bulbs will remain lit after 100 passes.

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:

Seven

Dividing the horses in a group of five, we will have five different groups of horses. A race will be conducted separately for each group which will help us find the fastest three horses in every group. We have used five different races till now.

Now we can have a race among the winners from every race which will help us determine the fastest of them all. We have used the sixth race.

The second fastest horse can be the one who came second in the sixth race or the second fastest horse in the finals winners' first race.

If we look at the first case, the third fastest horse can be the third fastest horse in the sixth race or the second fastest horse in either the fastest or second fastest horse first race.
If we look at the other case, the third fastest horse may be the second fastest in the sixth race or the third fastest in the final winners' first race.

If you calculate our assumptions, we have five more horses to race (seventh race) and determine the second fastest and the third fastest horse.

Solution:

Let us first assume that the number of drops required are N.

If the egg breaks at maximum number of tries, we will have N - 1 drops till it does not break. Thus we must drop the first egg from the height N. Now if the first drop of the first egg does not break the egg, we can have N - 2 drops for the second egg if the first egg breaks in the second drop.

Let us put that into a valid example for better understanding. Suppose we need 16 drops. Now let us drop the egg from the sixteenth floor, if it breaks, we will try all the floors below sixteen. Suppose it does not breaks, then we have 15 drops left and we will drop it from 16+15+1 = 32nd floor. This is because if it breaks at 32nd floor, we can try all the way down to 17th floor in fourteen tries making the total tries to be 16.

Let us assume that 16 is the correct answer

1 + 15 16 if breaks at 16 checks from 1 to 15 in 15 drops
1 + 14 31 if breaks at 31 checks from 17 to 30 in 14 drops
1 + 13 45.....
1 + 12 58
1 + 11 70
1 + 10 81
1 + 9 91
1 + 8 100 we can easily do in the end as we have enough drops to complete the task.

Seeking the above case, we must note that we have achieved it in 14 or 14 drops but we still need to find out the optimal one. We know from above that the optimal one will require 0 linear trials in the last step.

Thus we can say that

(1+p) + (1+ (p-1)) + (1+ (p-2)) + .........+ (1+0) >= 100.

Let 1+p=q which is the answer we are looking for

q (q+1)/2 >=100

q=14

Thus the optimal number of drops required are 14.

Drops will be tried at floor number 14, 27, 39, 50, 60, 69, 77, 84, 90, 95, 99, 100. How? Just move up 14, 13, 12 .... Floor till it breaks or does not.

Solution:

We know you must be having numerous thought about it. But trust us that when you know the solution to the problem you will regret for not thinking in that direction.

You just have to ask the person, "Which way is your city?" Just follow the path he directs you to and you will reach Truthcity. How?

Well if the person belongs to Liarcity, he will speak lie and will tell you the direction of the Truthcity and if the person belongs to Truthcity, he will tell the truth and will lead you to the Truthcity.

Isn't it simpler than you thought?

Solution:

He can ask any one of them. The question he will ask is - "Which door will the other guy say is the correct one?"

If he asks this question to the truth teller, he will get the incorrect door as he will honestly tell him what the liar will say. If he asks the liar, he will get the incorrect door again as he will lie about what the truth teller will tell him (the truth teller will obviously tell him the right door). Thus, by asking any of them, Sheldon Cooper will get the incorrect door only and he can choose the other one to find the treasure.

Solution:

Sherlock will ask any one of them "Are you the truth teller?" There can be two responses to this question - Yes or No.

If the person says yes, then he is the truth teller or a lying toggle.
He will ask the second question to the same person "Who is the truth teller?"
If he is the truth teller, he will admit it. If he is a lying toggler, he will speak truth on the second question if he lied on the first, telling you who the truth teller is.

If the person says no, Sherlock will be confirmed of the fact that he is not the truth teller and thus he is a toggle who has told truth and will say a lie on the second question.
The next question he will ask to the same person "Who is not the truth teller?"
Since the person told truth in the first question, he will speak a lie this time and in order to tell a lie, he will tell Sherlock exactly who the truth teller is.

Solution:

Everyone will appoint any one person as a leader. Now everyone must follow some steps. Every prisoner will flip the first switch up at the first opportunity and also at the second opportunity. If the first switch is already up or they have flipped it up two times already, then they will flip the second switch. The leader however will have a different routine. He is the only one who will flip the switch down the first switch. Suppose if the switch is already down, he will flip the second switch. Also he will keep a count and when he has flipped the first switch down 44 times, he will declare that every prisoner has visited the room.

This is because once the leader has flipped down the switch 44 times, he will know that everyone has visited the room. If the switch was down initially, the other 22 prisoners will turn it up two times. If the first switch was already up, there will be one prisoner who will flip the switch up only once and the rest will do it twice.

Now this trick won't work when the leader has flipped the switch 23 times. Consider the fact that it is possible that 12 prisoners randomly and the leader have been taken to the switch room 24 times and the rest have not been taken till then. Remember that the first switch might have been up in the beginning itself. Thus the prisoners must flip the switch up twice. If they do it just once, the leader will never know whether to count till 22 or 23.

Solution:

Christopher stole the necklace.

If we consider Billy's statements and the fact that one of the statements was a lie and the other was the truth, we can say that it must have been either Christopher or Eric.

Considering Dylan's statements and the same fact, it was either Christopher or Billy.

From both of these statements, we can deduce that it has to be Christopher.

Solution:

This question is one of the most common interview question. The most common answer of the question is sixty which is completely wrong. The second most common answer of this interview question is 10 which is wrong as well.

What we have is
6 fishermen catches 6 fishes in 6 minutes

If you offer them 10 times as much as time, they will surely catch 10 times as much as fishes as well
Thus the same six fishermen will be able to catch 60 fishes in 60 minutes.

You can also thing this in the unitary method.

If six fishermen catches 6 fishes in 6 minutes
Then, one fisherman will catch one 1 fish in 6 minutes
Thus in sixty minutes, one fisherman will catch 10 fishes (10 times the time, ten times the fishes).

Thus the right answer is six.