Solution:

5050

Explanation:
You must have noticed that 50 pairs of n^2 – (n-1) ^2 exist.
n^2 – (n-1)^2 = n + (n – 1)

In such case, we may as well write 100^2 - 99^2 + 98^2 - 97^2 + 96^2 - 95^2 + ... + 2^2 - 1^2 as 100 + 99 + 98+ ... + 2 + 1 = (100 x 101)/2 = 5050

Solution:

Think deeply about the door number 56, people will visit it for every divisor it has. So 56 has 1 & 56, 2 & 28, 4 & 14, 7 & 8. So on pass 1, the 1st person will open the door; pass 2, 2nd one will close it; pass 4, open; pass 7, close; pass 8, open; pass 14, close; pass 28, open; pass 56, close.
Thus we can say that the door will just end up back in its original state for each pair of divisor. But what about the cases in which the pair of divisor has analogous number for example door number 16? 16 has the divisors 1 & 16, 2 & 8, 4&4. But 4 is recurrent because 16 is a perfect square, so you will only visit door number 16, on pass 1, 2, 4, 8 and 16… leaving it open at the end. So only perfect square doors will remain open at the end.

Solution:

I am a Sponge

Solution:

Since the wife is widow, it is certain that the husband is dead. The law does not allow the dead person to marry.

Solution:

One of the cop will move clockwise while the other will move anti-clockwise every day. In such a manner, they will catch the thief in minimum seven days.

Solution:

Chalk Board

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:

This problem can be solved with similar triangles property.
The White triangle is similar to the Red one
Now Ratio of sides is 2:3 (as ratio of areas = ratio of based if the height is same)
Seeking that, White triangle = 9/4 * Yellow triangle = 9/2
Following all the above, the Green region = 5.5 units

Solution:

97

Explanation:

Difference between number sequence is below:
9,14,19,24,29

Now the difference between above sequence is 5.

So next number is 68 + 29 =>97

Solution:

Sun

Solution:

Too big too ignore

Solution:

Window

Solution:

I am a chicken in an egg

Solution:

Colors:
Y B R G
G R B Y
B Y G R
R G Y B


Signs:
x + - /
- / x +
/ - + x
+ x / -

Solution:

When you were inside the shed, the murderer replaced the lock with his own one that was identical to your original lock. When you locked the store room, you did not require a key and there was nothing abnormal for you. When you left, the murderer opened the lock with his key, planted the dead body inside and replaced the lock again putting the original lock in the place. He then closed it without any problem.

Solution:

8

Explanation:
If you look carefully then you will see that five matches lay ahead of the lighter and five are visible in the reflection of the lighter. Out of those five, we can see three matches emerging away from the surface of the lighter but two remains just as a part of reflection. Thus the total number of matches are eight.

Solution:

54325

Explanation:

Consider 1 = 5
2 = 25
3 = 325
4 = 4325
Then 5 = 54325

See the pattern. With every digit, the previous numbers are added after that digit

So the next number in the series is 54325

Solution:

Using Matchstick

Solution:

I am Water

Solution:

The woman told him to take out one nut each from the rest of the wheels and use them.