Solution:

The thief is the Pakistani guy. It is because the flag of Japan looks same when upside down and no one can tell if it is upside down. So, he was telling a lie.

Solution:

Sucre : 1 cupcake
T-Bag : 2 cupcakes
Burrows: 3 cupcakes
Scofield : 5 cupcakes

Explanation :
It's obvious if the prisoner has eaten 5 or more cupcakes than he will not ask this question "did you had more cupcakes than me?" as it will make him the person who had eaten maximum cupcakes.
A) Sucre : Must not have eater 5 cupcakes, therefore he has eaten anywhere between 1-4 cupcakes.

B) T-bag : Since sucre asked him the question T-bag knows sucre have eaten between 1 - 4 cupcakes, and therefore he must have eaten 2 to 4 cupcakes(it can't be 1 else he will reply answer as "No".)

C) Burrows : Replied with the answer as "I don't know and based on logic B) he knows T-bag must have eaten at least 2 and he replied no which implies that he must be eaten 3 or 4.

D) Scofield replied he know all, which indicates he must have eaten 5.
Therefore cupcake division is as below:
Sucre : 1
T-Bag : 2
Burrows: 3
Scofield : 5

Solution:

First Case:
The possible combinations of the adjacent bullets placed in the chambers are:
1, 2
2, 3
3, 4
4, 5
5, 6
6, 1

Now if you don't spin the barrel:
The first shot was empty and thus the combinations of (6, 1) and (1, 2) surely don't have bullets. This means that the bullets are present in some other possible 4 slots.

Therefore P (Death) = 1/4 = 0.25
P (Survival) = 3/4 = 0.75

If you chose to spin the barrel:
P (Death) = 2/6 = 1/3 = 0.33
P (Survival) = 4/6 = 2/3 = 0.77

Therefore you must not ask to spin the barrel and you will have a better chance at surviving.

Second Case:
If you don't spin it:
You were not hit in the first shot. Thus,
P (Death) in second hit = 2/5 = 0.4
P (Survival) in second hit = 3/5 = 0.6

If you spin the barrel
P (death) in the second hit = 2/6 = 1/3 = 0.33
P (Survival) in second hit = 4/6 = 2/3 = 0.77

Thus in this case, you have better survival rate if you chose to spin the barrel.

Solution:

This is because the queue for buying a donut is extremely long due to which the car engine becomes cool and gives trouble. While the queue for buying brownie is pretty small and thus the car gives no trouble at all.

Solution:

It is not as tough as it seems. The strategy is quite simple. You just have to pick up a card out of first three whenever you find a 9 or higher number.

Let us offer you a strong valid reason for the strategy. The probability that the four card kept below are below the number nine is 32*31*30*29 / 52*51*50*49 = 0.133
Thus in such case, you will lose $2 to $9 with equal probability = 0.133/8 = 0.0166

Let us now calculate the probability that one of the four cards is nine or even higher.
1 – 0.133 = 0.867
Now you will surely stop at the first sight of 9 or any value above 9 as per the strategy. Thus you have a chance of winning -1, 0, 1, 2, 3 with an equal probability = 0.867/5 = 0.173

In such case you stand a chance of winning overall amount of 0.14 every game you play.

Let us consider the other strategies. If you want to stop at the sight of 10 or above 10, the expected amount you will win per game will be 0.09 which is way below the present strategy. If you want to stop at above ten only, you will have an expected winning in negative.

Solution:

10 prisoners must sample the wine. Bonus points if you worked out a way to ensure than no more than 8 prisoners die.

Number all bottles using binary digits. Assign each prisoner to one of the binary flags. Prisoners must take a sip from each bottle where their binary flag is set.

Here is how you would find one poisoned bottle out of eight total bottles of wine.

Bottle 1 Bottle 2 Bottle 3 Bottle 4 Bottle 5 Bottle 6 Bottle 7 Bottle 8
Prisoner A X X X X
Prisoner B X X X X
Prisoner C X X X X
In the above example, if all prisoners die, bottle 8 is bad. If none die, bottle 1 is bad. If A & B dies, bottle 4 is bad.

With ten people there are 1024 unique combination so you could test up to 1024 bottles of wine.

Each of the ten prisoners will take a small sip from about 500 bottles. Each sip should take no longer than 30 seconds and should be a very small amount. Small sips not only leave more wine for guests. Small sips also avoid death by alcohol poisoning. As long as each prisoner is administered about a milliliter from each bottle, they will only consume the equivalent of about one bottle of wine each.

Each prisoner will have at least a fifty percent chance of living. There is only one binary combination where all prisoners must sip from the wine. If there are ten prisoners then there are ten more combination where all but one prisoner must sip from the wine. By avoiding these two types of combination you can ensure no more than 8 prisoners die.

One viewer felt that this solution was in flagrant contempt of restaurant etiquette. The emperor paid for this wine, so there should be no need to prove to the guests that wine is the same as the label. I am not even sure if ancient wine even came with labels affixed. However, it is true that after leaving the wine open for a day, that this medieval wine will taste more like vinegar than it ever did. C'est la vie.

Solution:

The murderer is Marcus.

The notes hinted towards the name of Marcus. How?
Just take the first letter from Compound, Art Room, Restroom, Underwater Room, Study Room and Movie Room and you will get:

C, A, R, U, S and M.

Rearranging these words leads to the name Marcus who is the murderer.

Solution:

What must be kept in mind before you start calculating through any method is that the weights can be put on both the pans.

So the weights - 1, 3, 7, 12, 43, 76, 102

The oldest man that can be weighed using these weights is a man aged 122 years

Solution:

The guy was killed a long ago, but he could not realize it till he reached his room. No need to be confused. Actually the killer has somehow inserted an extremely fine and thin incision blade in the belt of the guy. The guy did not feel anything at the moment as it was extremely fine. But when he was changing his clothes and took of his pant, the incision knife that was in his body, came out stuck to the belt cutting him very finely on the insides. That is the reason there was so much blood loss that he died and it was so fine that it was not visible to the naked eyes.

Solution:

The king had given them fake seeds and thus they would have never grown into a plant. While all the children had a flourishing plant in their pots, only one boy had an empty pot and thus only one of them was honest. So he announced him his heir.