Solution:

In the first wave, let us make an assumption that there is only 1 red hat and other 8 hats are blue. The student who can see all the other 8 blue hats on others' head will certainly be clear that there are 8 blue hats and 1 red hat since the professor told them that there is at least 1 red hat among the nine. But no one was able to answer after the first half an hour. Thus this can't be possible and our assumption is wrong.

Now let us assume that there are 2 red hats and 7 blue hats. The student who is having a red hat on his head can definitely see 7 blue hats and one red hat. He can deduce that he is wearing a red hat. But nobody was able to answer in the second interval as well. Thus our assumption is again wrong.

In the last wave of final five minutes, let us assume that there are 3 hats and 6 blue hats. The student with a red hat can see 2 other red hats and 6 blue hats and he may deduce that he is wearing a red hat. In this interval, everyone was able to answer which states that our assumption is correct.

Thus there are 3 red hats and 6 blue hats.

Solution:

146900

Explanation:
Whenever you face such type of questions, it is wise to begin from the last thing. Here in this question the last thing will be the 9th temple. He climbed down 100 steps and thus you know, he had Rs. 100 before beginning climbing down. Thus, he must have offered Rs. 100 to the god in that temple too (he offered half of the total amount). Also, he must have dropped Rs. 100 while climbing the steps of the ninth temple. This means that he had Rs. 300 before he begand climbing the steps of the ninth temple.

Now, we will calculate in the similar manner for each of the temples backwards.
Before the devotee climbed the eight temple: (300+100)*2 + 100 = 900
Before the devotee climbed the seventh temple: (900+100)*2 + 100 = 2100
Before the devotee climbed the Sixth temple: (2100+100)*2 + 100 = 4300
Before the devotee climbed the fifth temple: (4300+100)*2 + 100 = 8900
Before the devotee climbed the fourth temple: (8900+100)*2 + 100 = 18100
Before the devotee climbed the third temple: (18100+100)*2 + 100 = 36,500
Before the devotee climbed the second temple: (36500+100)*2 + 100 = 73300
Before the devotee climbed the first temple: (73300+100)*2 + 100 = 146900

Therefore, the devotee had Rs. 146900 with him initially.

Solution:

Tilt the barrel until the fruit juice barely touches the lip of the barrel. If the bottom of the barrel is visible then it is less than half full.If the barrel bottom is still completely covered by the fruit juice, then it is more than half full.

Solution:

Sansa has the siamese (conjoined) twin sister.

Solution:

The Lady sitting in the chair has the detector attached to the back of her blouse.

Solution:

There are 3 fruits (1 grapes, 1 apples, and 1 oranges).

Solution:

It can be solve as shown below.

Solution:

It can be down as shown below.

Solution:

99 matches.

The number of matches will always sum up to one less than the number of players in a knock out tournament. You may calculate it in any manner. Thus 99 matches were played.

Solution:

This problem can be solved with algebra as well as logical reasoning. We are going to tell you how it is logically possible.

Be it any quantity that was present in the beginning and any liquid as well. We know that we have taken a spoonful from one glass and put it into another. Then, we have taken a spoonful from the other glass and put it into the first glass. So, at the end of it, the quantity of liquid in either glass remains same as it was in the beginning.

Therefore, we can conclude the fact that any amount of coke that is missing in the glass with coke will be present in the lemonade glass. Also, the same quantity of lemonade will be missing from the lemonade glass and will be present in the coke glass.

Solution:

23rd day

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 = 276

Thus our anniversary falls on the 23rd day of the month.

You can apply other formulas to shorten the process but that is the simplest way to do it.