Question

Двоє друзів грають у гру «хрестики-нулики» на нескінченній площині у клітинку. Кожен намагається написати свій знак п'ять разів по вертикалі або по горизонталі. Доведіть, що той, хто розпочне другим, може зіграти так, що ніколи не програє. Решите пожалуйста дам 100 балов​

Answer 1

Ответ:

Пошаговое объяснение:

Когда один будет ставить в одну клетку крестик или нолик то получится так что не кто не сможет выйграть на бесконечной игре,это практически не возможно выйграть на бесконечной доске

Answer 2

The Game of "Tic-Tac-Toe" on an Infinite Plane

In the game of "Tic-Tac-Toe" (or "Xs and Os"), two players take turns placing their respective marks on a grid of cells. The objective is to create a line of five marks in a row, either horizontally, vertically, or diagonally. In this case, the game is played on an infinite plane, and each player aims to write their mark five times in a row either vertically or horizontally.

To prove that the player who starts second can always play in a way that ensures they never lose, we can use a strategy called "mirroring." This strategy involves the second player mirroring the moves of the first player, ensuring that they always have a valid move to make.

Let's break down the proof step by step:

1. The first player makes their initial move. They can choose any cell on the infinite plane to place their mark. 2. The second player mirrors the first player's move by placing their mark in the corresponding cell, which is symmetrically opposite to the first player's move. 3. From this point on, the second player continues to mirror the first player's moves. Whenever the first player places their mark in a new cell, the second player places their mark in the symmetrically opposite cell. 4. Since the game is played on an infinite plane, there will always be a symmetrically opposite cell for the second player to place their mark. 5. By mirroring the first player's moves, the second player ensures that they always have a valid move to make. They can simply place their mark in the symmetrically opposite cell. 6. Since the second player can always make a valid move, they can continue playing indefinitely without ever losing.

This strategy guarantees that the second player will never lose the game. They can always mirror the first player's moves and maintain a symmetrical position on the infinite plane.

Note: It's important to mention that this proof assumes both players are playing optimally and will not make any mistakes. If either player deviates from the optimal strategy, the outcome may differ.

I hope this explanation helps! Let me know if you have any further questions.