Alice marked several points on a line. Then she put more points – one point between each two adjacent points. Show that the total number of points on the line is always odd.
The Dormouse brought a \(4\times 10\) chocolate bar to share at the tea party. She needs to break the bar by the lines into single pieces (without any lines on it). In one turn she can cut one piece into two along the lines. What is the least number of cuts she needs to make to break the bar into single pieces?
Tweedledum and Tweedledee travel from their home to the castle of the White Queen. Having only one bicycle among them, they take turns in riding the bike. While one of them is cycling, the other one walks (walking means walking and not running). Nevertheless, they manage to arrive to the castle nearly twice as fast in comparison to if they both were walking all the way to the castle. How did they manage it?
The Hatter plays a computer game. There is a number on the screen, which every minute increases by 102. The initial number is 123. The Hatter can change the order of the digits of the number on the screen at any moment. His aim is to keep the number of the digits on the screen below four. Can he do it?
The March Hare decided to amuse himself by playing with three red and five blue sticks of various lengths. He noticed that the total length of all red sticks is equal to 30sm, and the total length of all blue sticks is equal to 30sm as well. Can he cut the sticks in such a way that every stick of one colour had a pair stick of the other colour of the same lenghs?
There are \(n\) inhabitants (\(n>3\)) in the Wonderland. Each habitant has a secret, which is known to him/her only. In a telephone conversation two inhabitants tell each other all the secrets they know. Show that after \((2n-4)\) conversations all the secrets may be spread among all the inhabitants.
The Hatter has 2016 white and 2017 black socks in his drawer. He takes two socks out of the drawer without looking. If the socks he takes out are of the same colour, he throws them away, and puts an additional black sock into the drawer. If the socks he takes out are of different colours, then he throws out the black sock, and puts the white one back. The Hatter continues with his sorting until there is only one sock left in the drawer. What colour is that sock?
The Hatter has a peculiar ancient device, which can perform the following three operations: for each \(x\) and \(y\) it calculates \(x+y\), \(x-y\) and \(\frac{1}{x}\) (for \(x \neq 0\)).
(a) The Hatter claims that he can square any positive real number using the device by performing not more than 6 operations. How can he do it?
(b) Moreover, the Hatter claims that he can multiply any two positive real numbers with the help of the device by performing not more than 20 operations. Can you show how?
(All intermediate results are allowed to be written down, and can be used in further calculations.)
Alice wants to mark 100 points on a plane by drawing it one by one, in such a way that no three points lay on one line, and at any moment while she marks the points down, the shape made up by the points has a symmetry line. Do you think it is possible?
One hundred and one numbers are written down: \(1^2\), \(2^2\), ..., \(101^2\). In one go it is allowed to erase any two numbers and write the absolute value of their difference instead. What is the smallest number which can be obtained as the result of 100 such operations?