We conclude the review of the book "Anticipate" with the following summary:
#Olympiad_problem
[Czech and Slovak Republics 1997]
Each side and diagonal of a regular n-gon (n ≥ 3) is colored blue or green. A move consists of choosing a vertex and switching the color of each segment incident to that vertex (from blue to green or vice versa). Prove that regardless of the initial coloring, it is possible to make the number of blue segments incident to each vertex even by following a sequence of moves. Also show that the final configuration obtained is uniquely determined by the initial coloring.
Let us take the example of a triangle. Since there are three vertices a, b and c in a triangle , any vertex has two edges incident on it. It doesn't have any diagonals. If we choose edges ab and ac to be blue and bc to be green as an initial configuration, then b and c don't have any even number of blues.
If we flip the color of edges incident on b, then ab becomes green and bc becomes blue, leading the colors of c to be even and blue.
If instead we flip the color of edges incident on c, the edges of b become both blue.
Since a already had even number of blue edges to begin with, all the vertices of the triangle have each been able to make their incident edges blue.
The total number of edges in a triangle is 3 and because its an odd number, the blues and greens are divided unequally. Therefore permitting them to switch the majority with each move. Futhermore the majority can be even. Lastly all the moves at a vertex can be the same for all the vertices since they are all the same. But the initial coloring configuration is unique and therefore the flipping to make the coloring at a vertex even and blue is also unique.
The final part is about our visionary self and how to improve it.
This section asks you to begin by trying to write your own obituary so that we can gain clarity about our own beliefs and values as a critical step. Another exercise in this way would be to ask a friend to interview you and respond with honest answers. Questions could include say, describe three different situations in which you were truly at the best.
Then the section asks us to lead by example. Being mindful of these values helps us observe what may have been overlooked. Another practice is to deliberately break your normal pattern of working, communicating, thinking, reacting or responding. Get radical exposure by engaging with folks who are different from us. And finally try opinion swap such as adopting the opinion of someone you don’t necessarily agree with.
This is where we use the power of language. We use verbs to carry sentences. We can use a predefined list of verbs that attract attention. Similarly show picture. This evokes imagination and helps them envision a different world. Also metaphors make the message stick. If you give an example with a metaphor, it’s much easy to relate to. Likewise analogies can be drawn to something that is actionable. If you say you want something to be the ivy league in its industry, it becomes clearer. Finally, stories inspire. They are catchy and heighten our natural curiosity.
#Olympiad_problem
[Czech and Slovak Republics 1997]
Each side and diagonal of a regular n-gon (n ≥ 3) is colored blue or green. A move consists of choosing a vertex and switching the color of each segment incident to that vertex (from blue to green or vice versa). Prove that regardless of the initial coloring, it is possible to make the number of blue segments incident to each vertex even by following a sequence of moves. Also show that the final configuration obtained is uniquely determined by the initial coloring.
Let us take the example of a triangle. Since there are three vertices a, b and c in a triangle , any vertex has two edges incident on it. It doesn't have any diagonals. If we choose edges ab and ac to be blue and bc to be green as an initial configuration, then b and c don't have any even number of blues.
If we flip the color of edges incident on b, then ab becomes green and bc becomes blue, leading the colors of c to be even and blue.
If instead we flip the color of edges incident on c, the edges of b become both blue.
Since a already had even number of blue edges to begin with, all the vertices of the triangle have each been able to make their incident edges blue.
The total number of edges in a triangle is 3 and because its an odd number, the blues and greens are divided unequally. Therefore permitting them to switch the majority with each move. Futhermore the majority can be even. Lastly all the moves at a vertex can be the same for all the vertices since they are all the same. But the initial coloring configuration is unique and therefore the flipping to make the coloring at a vertex even and blue is also unique.
No comments:
Post a Comment