Question 14
João has three-color buttons: blue, white and gray. We will represent a blue button by $A$, a white button by $B$ and a gray button by $C$. It places five buttons on the vertices of a regular pentagon clockwise: $A, C, B, C$ and $B$, as in the first figure below. He knows he can do some button changes by following a certain rule. When choosing a button, it can exchange it for one of another color as long as there are no two buttons with the same color in neighboring vertices. In order not to change buttons without objective, he decides to adopt as a goal to leave the buttons with the colors $A, B, C, B$ and $C$ clockwise as indicated in the second figure below. 
After making several changes, João was unable to reach the desired final situation. He then decided to ask his teacher for help from Piraldo. The teacher states that it is not possible to start from the initial configuration and reach the desired final configuration. To prove this, Professor Piraldo took some cards with the numbers $1,-1$ and 0 . Then he told John to put a card on each side of the Pentagon with the following pattern: i) Looking clockwise, if the vertices have $B$ and $A$ or have $A$ and $C$ Place a card with the number 1 . ii) Still looking clockwise, if the vertices have $A$ and $B$ or have $C$ and $A$, place a letter with the number -1 . iii) For other cases, place the letter 0 . Whenever you make an exchange, change the cards using these instructions. After the explanation, Professor Piraldo said that analyzing the behavior of the letters João could now conclude because it is impossible to achieve the initial objective.
a) doing all possible button changes, from $A$ to $B$, from $B$ to $C$ or $C$ for $A$ and reverse exchanges, what happens to the sum of the cards placed on the sides according to Professor Piraldo's instructions?
b) What is the sum of the cards in the initial situation and the desired final situation? Complete that it is not possible to start from the initial configuration and reach the desired final situation.
After making several changes, João was unable to reach the desired final situation. He then decided to ask his teacher for help from Piraldo. The teacher states that it is not possible to start from the initial configuration and reach the desired final configuration. To prove this, Professor Piraldo took some cards with the numbers $1,-1$ and 0 . Then he told John to put a card on each side of the Pentagon with the following pattern: i) Looking clockwise, if the vertices have $B$ and $A$ or have $A$ and $C$ Place a card with the number 1 . ii) Still looking clockwise, if the vertices have $A$ and $B$ or have $C$ and $A$, place a letter with the number -1 . iii) For other cases, place the letter 0 . Whenever you make an exchange, change the cards using these instructions. After the explanation, Professor Piraldo said that analyzing the behavior of the letters João could now conclude because it is impossible to achieve the initial objective. a) doing all possible button changes, from $A$ to $B$, from $B$ to $C$ or $C$ for $A$ and reverse exchanges, what happens to the sum of the cards placed on the sides according to Professor Piraldo's instructions?
b) What is the sum of the cards in the initial situation and the desired final situation? Complete that it is not possible to start from the initial configuration and reach the desired final situation.