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 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 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?
Two pirates Bob and John were boasting that they could make the strongest coctail. Bob mixed together rum and gin, and John mixed vodka and port. It is known that rum is stronger than vodka, and gin is stronger than port. Can it be that John’s drink was stronger than Bob’s?
Is it possible that the sum and the product of some given natural numbers is equal to 99?
Selena wrote down some positive numbers. She added up those numbers, and the resulting sum was greater than 10. Then she decided to add up the squares of those numbers. Could it be possible that the sum of the squares of the numbers was less than 0.1?