 Na imagem visualizamos o logotipo do projeto 'MatematiZou', escrito com letras de forma na cor verde-azulada. A letra “Z” está na cor branca dentro de um hexágono com pontas arredondadas preenchido na mesma tonalidade de cor do restante das letras.  Na imagem temos um mini caminhão branco de madeira, cuja caçamba é azul escura e está escrito em letras brancas “UFABC”. As rodas do caminhão são vermelhas e quadradas. Cada par de rodas do caminhão está em cima de uma pista com ondulações.  Na imagem, há várias pessoas participando do evento “UFABC para todos” no Ginásio da UFABC campus Santo André. Algumas pessoas estão ao redor de uma mesa com jogos que incluem copos, mapas, tabuleiro de xadrez, entre outros, e outras, estão vendo as exposições espalhadas por todo o ambiente.](https://matematizou.gradmat.ufabc.edu.br/files/LogoMatematizouTeal.png)
Historic context
This game was presented as a problem in math olympiads and aimed to describe a strategy that ensures the first player will not lose. The game has simple rules, and in the version presented here, all you need to know is how to make sums to play. The solution to this problem involves basic mathematical skills and simple concepts like parity.
Game’s description
This is a game that involves two players and a box of coins (or tokens) marked with random values.
- Before starting:
Someone takes an even number of coins from the large pile (randomly) and places them on the table in a row.
Both players will have access to the following informations before starting the game:
(a) The values of each of the coins;
(b) The order in which the coins are shown. - How to play:
In turns, each player has to choose one of the two coins at the end of the row and keep it. At this point, the next player will have to do the same with the new row of coins, but with one coin less. (The only thing that changes in each turn is one of the ends, there is no shuffling!).
This goes on until there are no more coins on the table (each player has half of the coins in the end of the game). - Goal:
Whoever has the coins with the greatest sum value in the end, wins!
Example
In this example, the strategy that ensures the victory of Player 1 will be used!
To make it easier, when we place a number in red, it means that the corresponding coin has already been taken out of the game!
Starting coins configuration:
1st round:
Player 1’s coins: \{3\}
Player 2’s coins:
Jogada 2:
Player 1’s coins: \{3\}
Player 2’s coins: \{6\}
Jogada 3:
Player 1’s coins: \{3,2\}
Player 2’s coins: \{6\}
Jogada 4:
Player 1’s coins: \{3,2\}
Player 2’s coins: \{6,1\}
Jogada 5:
Player 1’s coins: \{3,2,4\}
Player 2’s coins: \{6,1\}
Jogada 6:
Player 1’s coins: \{3,2,4\}
Player 2’s coins: \{6,1,0\}
Jogada 7:
Player 1’s coins: \{3,2,4,-2\}
Player 2’s coins: \{6,1,0\}
Jogada 8:
Player 1’s coins: \{3,2,4,-2\}
Player 2’s coins: \{6,1,0,-1\}
Winner: Player 1
How can Player 1 always assure his victory? What is the winning strategy?
