How do you cut a rectangular cake into two equal pieces when someone has already removed a rectangular piece from it? The removed piece can be of any size or orientation. You are allowed just one straight cut.
Why does a mirror reverse right and left instead of up and down?
Max asked Emily how old she was. She replied that she was 13 years old the day before yesterday, and will be 16 next year. Then, Max asked her brother, whether it was true, and he said yes. How is it possible if nobody was lying?
Place coins on a \(6\times 6\) chequered board (one coin on one square), so that all the horizontal lines contain different number of coins, and all vertical lines contain the same number of coins.
A young and ambitious software engineer is working on his own basic version of an intelligent personal assistant. The application can only answer closed questions (a closed question is a question that can be answered only ‘yes’ or ‘no’). He installs this application on three mobile devices and runs a set of tests. He discovers there is one unstable device. From time to time the application gives wrong answers, but you cannot really predict when. Being exhausted after unsuccessful attempts to find the mistake in his code, the software engineer goes to sleep. The next morning he cannot remember which device is not working properly. Taking into account that devices are connected to the same server (so normally working applications can detect which one is not always receiving the signal) explain how in two questions the engineer can determine the unstable device. One question is for one device only.
A pencil box contains pencils of different colours and different lengths. Show that it is possible to choose two pencils of both different colours and different lengths.
The text for this problem was originally typed in three different fonts and in three different colours. The original style is lost now, and Bella and Louise disagree on the following. Bella says that whatever the original font was it was always possible to choose three letters from the text such that all the three colours and all the three fonts were presented in that triple, and Louise does not think so. Who is right?
a) A bachelor student Peter haven’t slept properly for the last month. One of the reasons for that among many others was that every Monday at 3 p.m. he had a deadline for submitting his weekly calculus assignments. During the first month he counted six deadlines. Can it be the case or would you advise him to have more sleep?
(b) Once Peter checked the table with the assignment results he realized there were fewer Mondays in the last month. Is it possible there were only five Mondays?
Scrooge McDuck has 100 golden coins on his office table. He wants to distribute them into 10 piles so that no two piles contain the same amount of coins. And moreover, no matter how you divide any of the piles into two smaller piles among the resulting 11 piles there will be two with the same amount of coins. Sounds impossible? Try to find a suitable example. Scrooge spent a while on working out this question, maybe he will even give you a penny.
There are 36 parcels weighing 1 kg, 2 kg, 3 kg, ..., 36 kg. Today only three cars are in service. Each car has a capacity of 12 parcels. Can one distribute all packages between the cars in such a way that each vehicle has the same total weight of parcels?