A natural number \(n\) can be exchanged to number \(ab\), if \(a+b=n\) and \(a\) and \(b\) are natural numbers. Is it possible to receive 2017 from 22 after such manipulations?
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?
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?
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?
One of your employees insists on being paid daily in gold. You have a gold bar whose value is that of seven days’ salary for this employee. The bar is already segmented into seven equal pieces. If you are allowed to make just two cuts in the bar, and must settle with the employee at the end of each day, how do you do it?
You have a 3-quart bucket, a 5-quart bucket, and an infinite supply of water. How can you measure out exactly 4 quarts?
Suppose you had eight billiard balls, the recruiter began. One of them is slightly heavier, but the only way to tell is by put-ting it on a scale against the others. What’s the fewest number of times you’d have to use the scale to find the heavier ball?