Question

Двоє гравців по черзі дістають з скриньки кульки.Програє той,хто забирає останню кульку.Хто може забезпечити собі перемогу,перший чи другий,Якщо спочаткув скриньці було 2002 кульки і за один хід можна виймати не менше однієї і не більше пяти кульок?

Answer 1

Другий звичайно переможе думав 9 хвилин я

Answer 2

The Game of Removing Balls from a Box

In this game, two players take turns removing balls from a box. The player who takes the last ball loses the game. The question is, who can guarantee a win, the first player or the second player? Let's analyze the situation.

The Initial Number of Balls

At the beginning of the game, there are 2002 balls in the box. Each player can remove a minimum of one ball and a maximum of five balls in a single turn.

Analyzing the Game

To determine who can secure a victory, we need to analyze the game and identify any winning strategies.

The Winning Strategy

The key to winning this game is to understand the concept of "winning position" and "losing position." A winning position is one where, regardless of the opponent's move, the player can always force the game into a losing position for the opponent. A losing position is one where, regardless of the player's move, the opponent can always force the game into a winning position for themselves.

Applying the Strategy

In this particular game, we can determine the winning strategy by analyzing the number of balls in the box. Let's consider the possible scenarios:

1. If there are 1 to 6 balls in the box, the first player can take all the balls and win the game. 2. If there are 7 to 12 balls in the box, the first player can take enough balls to leave the second player with 6 balls. From this point, the second player can always mirror the first player's moves and force the game into a winning position for themselves. 3. If there are 13 to 18 balls in the box, the first player can take enough balls to leave the second player with 12 balls. Again, the second player can mirror the first player's moves and secure a win. 4. This pattern continues, and we can observe that the winning positions repeat every 6 balls. Therefore, if the number of balls in the box is a multiple of 6, the second player can always force a win.

Applying the Strategy to the Given Scenario

In the given scenario, there are 2002 balls in the box. Since 2002 is not a multiple of 6, the first player can always force a win by applying the winning strategy described above.

Conclusion

In the game where two players take turns removing balls from a box, the first player can guarantee a win if the number of balls in the box is not a multiple of 6. In the given scenario with 2002 balls, the first player can secure a victory.

[[1]]