In the context of Example 14.2 what is the answer if we have five numbers instead of four? (i.e., can we get four distinct prime numbers then?)
Now George is sure he found two numbers with the quotient equal to their sum. And on top of that their product is still equal to the same value. Can it be true?
A maths teacher draws a number of circles on a piece of paper. When she shows this piece of paper to the young mathematician, he claims he can see only five circles. The maths teacher agrees. But when she shows the same piece of paper to another young mathematician, he says that there are exactly eight circles. The teacher confirms that this answer is also correct. How is that possible and how many circles did she originally draw on that piece of paper?
A group of three smugglers is offered to smuggle a chest full of treasures across the dangerous river. The boat they possess is old and frail. It can carry three smugglers without the chest, or it can carry the chest and only two smugglers. The price for this job is extremely high, and the gang is more than interested in completing the job. Think of a strategy the smugglers should follow to successfully transit the chest and themselves to the other shore.
It is easy to construct one equilateral triangle from three identical matches. Can we make four equilateral triangles by adding just three more matches identical to the original ones?
My mum once told me the following story: she was walking home late at night after sitting in the pub with her friends. She was then surrounded by a group of unfriendly looking people. They demanded: “money or your life?!” She was forced to give them her purse. She valued her life more, since she was pregnant with me at that time. According to her story she gave them two purses and two coins. Moreover, she claimed that one purse contained twice as many coins as the other purse. Immediately, I thought that the mum must have made a mistake or could not recall the details because of the shock and the amount of time that passed after that moment. But then I figured out how this could be possible. Can you?
(a) In a regular 10-gon we draw all possible diagonals. How many line segments are drawn? How many diagonals?
(b) Same questions for a regular 100-gon.
(c) Same questions for an arbitrary convex 100-gon.
A hedge fund is intending to buy 50 computers and connect each of them with eight other computers with a cable. Please do not ask why they need to do that, that is a top secret never to be made public! A friend of mine said that it’s related to some cryptocurrency research, but you should immediately forget all I just told you; it would be unwise to spread rumours! Let’s go back to the mathematical part of this story and stop the unrelated talk. The question is, how many cables do they need?
At a party there are people dressed in either blue or green. Every person dressed in blue had a chance to dance with exactly \(7\) people in green, only once with each one. On the other hand, every person in green danced exactly with \(9\) people in blue, also only once with each. Were there more people dressed in blue or in green at the party?
Draw \(6\) points on a plane and join some of them with edges so that every point is joined with exactly \(4\) other points.