a) Twelve inhabitants are sitting under a big tree. One of them says, “There is at least one liar among us”. How many knights can be among them?
b) A different group of other twelve inhabitants is resting by the river. Suddenly, one of them exclaims, “If everyone except me is a liar, then I am a liar too”. How many liars can be among them?
Ten inhabitants are sitting around the table. Each one of them says, “My neighbour on the right is a knight”. How many knights and liars are sitting there?
lbert (A), his wife Beatrix (B), and their children Charlie (C), Dan (D) and Elizabeth (E) live in a bungalow. They have a really nice TV set. It is known that
1) If A is watching the TV, then B is watching the TV.
2) At least one of D and E is watching the TV.
3) Only one of B and C is watching the TV.
4) Either C and D are watching the TV together, or both are not watching.
5) If E is watching the TV, then both A and D are also watching the TV. Can you tell who is watching the TV in this family and who is not?
Once I found a really strange notebook. There were 100 statements in the notebook, namely
“There is exactly one false statement in this notebook.”
“There are exactly two false statements in this notebook.”
“There are exactly three false statements in this notebook.”
...
“There are exactly one hundred false statements in this notebooks.”
Are there any true statements in this notebook? If there are some true statements, then which ones are true?
There are some coins laying flat on the table, each with a head side and a tail side. Fifteen of them are heads up, the others are tails up. You can’t feel, see or in any other way find out which side is up, but you can turn them upside down. Split the coins into two piles such that there is the same number of heads in each pile.
For an experiment a researcher puts a dot of invisible ink on a piece of paper and also draws a square with regular ink on the paper. In the experiment, a subject will draw a visible straight line on the page and the researcher, who has on special eyeglasses for spotting the dot, will tell the subject which side of the line the dot of invisible ink is on. If the dot is on the line, the researcher will tell the subject it is on the line. What is the smallest number of straight lines the subject needs to draw to figure out for sure whether the invisible dot lies in the square?
Can a \(5\times5\) square checkerboard be covered by \(1\times2\) dominoes?
Can a knight start at square a1 of a chessboard, and go to square h8, visiting each of the remaining squares exactly once on the way?
The King and Knave of Hearts were playing a game of croquet. The Knave of Hearts went first and made a sensational hit that created a closed trajectory of 9 line segments. It is now the King’s turn and he is worried that he cannot possibly match the same sensational hit of the Knave’s move. Can he be lucky enough to cross all the 9 segments (of the hit created by the Knave) with one straight hit not passing through the vertices?
The Queen has introduced a new currency in the world of Wonderland. This currency consists of three golden coins with values \(3\), \(5\) and \(15\). Is it possible for Alice to change an old note with value \(100\) using \(11\) new coins?