Your friend pulls out a perfectly circular table and a sack of quarters, and proposes a game.
"We'll take turns putting a quarter on the table," he says. "Each quarter must lay flat on the table, and cannot sit on top of any other quarters. The last person to successfully put a quarter on the table wins."
He gives you the choice to go first or second. What should you do, and what should your strategy be to win?

You should go first, and put a quarter at the exact center of the table.
Then, each time your opponent places a quarter down, you should place your next quarter in the symmetric position on the opposite side of the table.
This will ensure that you always have a place to set down our quarter, and eventually your oppponent will run out of space.

A man was to be sentenced, and the judge told him, "You may make a statement. If it is true, I'll sentence you to four years in prison. If it is false, I'll sentence you to six years in prison." After the man made his statement, the judge decided to let him go free.What did the man say?

He said, "You'll sentence me to six years in prison." If it was true, then the judge would have to make it false by sentencing him to four years. If it was false, then he would have to give him six years, which would make it true. Rather than contradict his own word, the judge set the man free.

Last week, the local Primary school was visited by the Government School Inspector who was there to check that teachers were performing well in their respective classes. He was very impressed with one particular teacher. The Inspector noticed that each time the class teacher asked a question, every child in the class put up their hands enthusiastically to answer it. More surprisingly, whilst the teacher chose a different child to answer the questions each time, the answers were always correct.
Why would this be?

The children were instructed to ALL raise their hands whenever a question was asked. It did not matter whether they knew the answer or not. If they did not know the answer, however, they would raise their LEFT hand. If they knew the answer, they would raise their RIGHT hand. The class teacher would choose a different child each time, but always the ones who had their RIGHT hand raised.

It was a grandeur party. In order to filter the uninvited guests, the security guard was assigned a task to check the secret password. The guests invited by the royal family also were shared with the secret password.
John wasn't an invited guest. He learned that the password is needed to make an entry. He hides himself and started watching the guests and the security.
The first guest comes. Security told him, TWELVE and the guest replied SIX. He wished him and allowed him to enter.
The second guest comes. Security told him SIX and the guest replied THREE! He was too allowed.
John made an entry as third guest. Security told him EIGHT and John replied FOUR. He was thrown out of the party!
Why?

The answer should be five. The password is not half of the digit, but the number that represents the number of digits told by security.

On the first day they cover one quarter of the total distance.
The next day they cover one quarter of what is left.
The following day they cover two fifths of the remainder and on the fourth day half of the remaining distance.
The group now have 14 miles left, how many miles have they walked?

In the land of Brainopia, there are three races of people: Mikkos, who tell the truth all the time, Kikkos, who always tell lies, and Zikkos, who tell alternate false and true statements, in which the order is not known (i.e. true, false, true or false, true, false). When interviewing three Brainopians, a foreigner received the following statements:
Person 1:
I am a Mikko.
Person 2:
I am a Kikko.
Person 3:
a. They are both lying.
b. I am a Zikko.
Can you help the very confused foreigner determine who is who, assuming each person represents a different race?

Person 1 is a Miko.
Person 2 is a Ziko.
Person 3 is a Kikko.

You can easily "tile" an 8x8 chessboard with 32 2x1 tiles, meaning that you can place these 32 tiles on the board and cover every square.
But if you take away two opposite corners from the chessboard, it becomes impossible to tile this new 62-square board.
Can you explain why tiling this board isn't possible?

Color in the chessboard, alternating with red and blue tiles. Then color all of your tiles half red and half blue. Whenever you place a tile down, you can always make it so that the red part of the tile is on a red square and the blue part of the tile is on the blue square.
Since you'll need to place 31 tiles on the board (to cover the 62 squares), you would have to be able to cover 31 red squares and 31 blue squares. But when you took away the two corners, you can see that you are taking away two red spaces, leaving 30 red squares and 32 blue squares. There is no way to cover 30 red squares and 32 blue squares with the 31 tiles, since these tiles can only cover 31 red squares and 31 blue squares, and thus, tiling this board is not possible.