John is a knight, he never lies. But when you ask him the same question twice, his second answer suddenly is different from the first. How is it possible?
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?
The product of 22 integers is equal to 1. Show that their sum cannot be zero.
A bus, a truck, and a motorcycle move without acceleration and pass a stationary observer at equal time intervals in the order mentioned above. They pass another observer farther down the road at the same equal time intervals but in different order. This time the order is the bus, the motorcycle, the truck. Find the speed of the bus, if the speed of the truck is 30 mph, and the speed of the motorcycle is 60 mph.
The area of a rectangle is 1 cm\(^2\). Can its perimeter be greater than 1 km?