Hard Riddles

If there is only one person with grey colored eyes, he will see all other people and find out that no one has grey eyes. Thus he will come to know that he himself has grey eyes. Thus, he will leave the island that very day.

If there are two people with grey eyes, they will find only one pair of grey eyes and will think that that is the person with grey eyes. So they both won't leave the island. On the next day however, they will see that no one left the island and they will understand that both of them has the same eye color that is grey color.

Now through induction, since there are 100 people with grey eyes, all of them will leave the island on the 100th day after the announcement was made by the outsider.

This is because the queue for the chocolate ice cream is very long. Thus, the engine gets cold and becomes troublesome. While the queue for strawberry ice cream is short.

C is the top view of the given pyramid.

Total number of matches that are to be played are:
9C2 * 2
= 9! / (2!7!) * 2
= 8 * 9
= 72
And thus each team will play 16 matches.

Now let us examine the possible cases.

Case 1:
If suppose, Team A, Team B and Team C wins 14 matches each i.e. they lose 1 match each from the other two teams, then the Team D can win only 16 - 3 * 2 = 10 matches.

Case 2:
Now let us suppose if the top four teams win 13 matches each.
Team A: 13 wins and 3 loses from Team B, Team C and Team D.
Team B: 13 wins and 3 loses from Team A, Team C and Team D.
Team C: 13 wins and 3 loses from Team A, Team B and Team D.
Team D: 13 wins and 3 loses from Team A, Team B and Team C.

In this case, the Team E can only win a maximum of 8 matches as Team A, Team B, Team C and Team D will win all the matches played by E.

Case 3:
If suppose the top five team wins 12 matches each.
Team A: 12 wins and 3 loses from Team B, Team C, Team D and Team E.
Team B: 12 wins and 3 loses from Team A, Team C, Team D and Team E.
Team C: 12 wins and 3 loses from Team A, Team B, Team D and Team E.
Team D: 12 wins and 3 loses from Team A, Team B, Team C and Team E.
Team E: 12 wins and 3 loses from Team A, Team B, Team C and Team D.

Now Team E wins 12 matches but it still won't be able to qualify for the playoffs (assuming that it has the lowest run rate).

Therefore from case 2, it can be said that a team must win at least 13 matches to be sure to play in the playoffs.

There are 12 squares in the picture.

The trick to find it would be drawing diagonals. We know that the sides of a square are parallel and thus, the diagonals should be parallel as well. Let us begin with the outer most rectangle. See the picture. The yellow lines are diagonals drawn for reference.

http://1.bp.blogspot.com/-aGHzJWfxtEY/U2_Yh0HBt1I/AAAAAAAAC6s/VCeYXzdT1rk/s1600/how-many-squares-puzzle-solution-1.png

You can make out four squares in the image.

http://1.bp.blogspot.com/-lTktLIzp260/U2_Yh67sYFI/AAAAAAAAC68/JHsFRRPVhMs/s1600/how-many-squares-puzzle-solution-2.png

Now see the next image. You can make out a total of 8 squares - 3 blue, 3 green and 2 red. There are 4 other rectangles in the figure but they cant be square as their diagonal is not parallel with the yellow lines we drew for reference.

Thus, total squares = 4 + 8 = 12.

Let us say that the three people are A, B and C respectively. Now you, A will tell the sum of your salary and a random number to B.
SA (Actual Salary) + A (random number of A)

In the same manner your friend B, will add his salary with a random number to the amount given by A (you) and will pass it to C.
(SA + A) + (SB + B)

C will do the same.
(SA + A) + (SB + B) + (SC + C)

Now C will pass this total amount to A. He will subtract his random number from the amount and it will become:
(SA) + (SB + B) + (SC + C)

Then he will pass it to B and B will subtract his random number from the amount.
(SA) + (SB) + (SC + C)

Then he will pass this to C who will subtract his random number.
(SA) + (SB) + (SC)

Now C has the total amount of their combined salaries. He will tell the amount to all of you and you can easily divide it by 3 to get the average salary.

NOWHERE

The two words hinted are:
NOW (The present time)
HERE (The present place)

Combining them, we get NOWHERE (A lack of place).

Analyzing thing, the friend has better chances of winning. Let us assume and analyze what can happen at any point of time supposing that we have not already completed the sequence. If the last flip was tails, start at 1 and if it is heads, start at 2.

1) If any one between us flips to get tails, nothing will change. We will still be at step 1 and we both will be needing 3 additional flips to complete. But if either of us gets heads, we continue.

2) If either of us gets heads, we stay on this step. But if we get tails, we continue. At this point, I am 3A and my friend is 3B.

3A) If I get tails now, I will have to start all over again from step 1 till I again get heads and I will have to flip the coin at least 3 times to complete my sequence.

3B) If the friend gets heads, he goes back to the step 2 and try again for tails. He will have to flip the coin at least 2 times to complete his sequence.

Thus, he obviously has better chances of winning.

29

The center number is the product of the largest cornering numbers minus the square of the smallest one.

Thus the number is:
45 – 16 = 29

P and Q cross the bridge first taking two minutes.
P comes back with which makes the total time three minutes.
R and S cross this time making the total time thirteen minutes.
Q comes back with the torch making the total time fifteen minutes.
Now P and Q cross the bridge finally taking two more minutes and making the total time seventeen minutes.

Thus all were able to reach before the bridge collapsed.