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?
One sunny day Alice met the White Rabbit. The Rabbit told her that he owns a pocket watch which has 11 gears arranged in a chain loop. The rabbit asked Alice if it was possible for all the gears to rotate simultaneously. What is your opinion on this matter? Can all the gears rotate simultaneously?
After the Mad Tea-Party, the Hatter was so excited that he decided to cool down by going on a short walk across the chessboard. He started at position a1, then walked around in steps taking each step as if he was a knight, and eventually returned back to a1. Show that he made an even number of steps.
Is it possible that odd integers \(a\), \(b\), \(c\), \(d\) satisfy \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=1\)?
The Cheshire Cat wrote one of the numbers \(1, 2,\dots, 15\) into each box of a \(15\times15\) square table in such a way, that boxes which are symmetric to the main diagonal contain equal numbers. Every row and column consists of 15 different numbers. Show that no two numbers along the main diagonal are the same.
The product of 22 integers is equal to 1. Show that their sum cannot be zero.