Tough Decisions

You are driving your car along the road in a very harsh snowy weather and reach a bus stop. On this bus stop you see that there are three people waiting – your best friend, a sick old lady and the girl of your dreams. The car unfortunately can accommodate only 2 people (including you), so you can not take all of them with you. What will be your choice?

The best solution is to let your friend drive the old lady and you stay with the girl of your dreams on the bus stop.

Optical Illusions

If you count carefully the number of people before the tiles scramble and after that, you will see that one person disappears. Can you explain how this is possible?

Similarly, on this picture it looks like after changing the places of the tiles on the diagram, their total area decreases by one. Can you explain this?

If you look carefully, you will notice that every person on the picture with 12 people is slightly taller than his corresponding person on the picture with 13 people. Basically, we can cut little pieces from 12 different people without making noticeable changes and arrange them into a new person.

For the second question, none of the shapes before and after the scrambling is really a triangle. One of them is a little bit curved in at the hypothenuse and at the other one is a little bit curved out. This is barely noticeable, because the red and the blue triangle have very similar proportions of their sides – 5/2 ~ 7/3.

Friends and Enemies

Show that in each group of 6 people, there are either 3 who know each other, or 3 who do not know each other.

Let’s call the people A, B, C, D, E, F. Person A either knows at least 3 among B, C, D, E, F, or does not know at least 3 among B, C, D, E, F.

Assume the first possibility – A knows B, C, D. If B and C know each other, C and D know each other, or B and D know each other, then we find a group of 3 people who know each other. Otherwise, B, C, and D form a group in which no-one knows the others.

If A doesn’t know at least 3 among B, C, D, E, F, the arguments are the same.

Scoring penalties

At some point in Leonel Messi’s career, the football player had less than 80% success when performing penalty kicks. Later in his career, he had more than 80% success when performing penalty kicks. Show that there was a moment in Leonel Messi’s career when he had exactly 80% success when performing penalty kicks.

Let us see that it is impossible for Messi to jump from under 80% success rate to over 80% success rate in just one attempt. Indeed, if Messi’s success rate was below 80% after N attempts, then he scored at most 4N/5 – 1/5 = (4N-1)/5 times. If his success rate was above 80% after N+1 attempts, then he scored at least 4(N+1)/5 + 1/5 = (4N-1)/5 + 6/5 times. However, Messi can not score more than one goal in a single attempt, which completes the proof.