Find a non-intersecting closed path, passing through all four ants, which traverses this double-sided maze. You can not switch between the top and the bottom side of the path at no times.
The solution is shown below.
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.
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
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.
How can a baby fall out of a 30-story building onto the ground and still be alive?
The baby fell from the first floor.
Clearly, Mick should not aim for Rick, because if he kills him, then he will be killed by Nick. Similarly, Nick should not aim for Mick, because if he kills him, then he also will be killed by Nick. Therefore, if Nick ends up against alive Mick and Rick, he will aim at Rick, because would prefer to face off a weaker
Now if Mick shoots at Nick and kills him, then he will have to face off Rick with chance of survival less than 1/3. Instead, if he decides to shoot in the air, then he will face off Rick or Nick with chance of survival at least 1/3. Therefore Mick’s strategy is to keep shooting in the air, until he ends up alone against one of his opponents.
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