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?
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?
You have two sticks and matchbox. Each stick takes exactly an hour to burn from one end to the other. The sticks are not identical and do not burn at a constant rate. As a result, two equal lengths of the stick would not necessarily burn in the same amount of time. How would you measure exactly 45 minutes by burning these sticks?
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?
How many times a day do a clock’s hands overlap?