Question

Consider the survival game below:

In this game, the playing field consists of numbers from 0 to d-1 (0,1, 2, ..., d-2, d-1), and the players take turns making moves according to their respective rules, with all moves being performed modulo d:
-Batu can add 1 or 2 or 3 to the current position (mod d)
-Çağatay can multiply with one of 2, 3, 5, 7 (mod d)
The game starts with Batu making the first move from position 2, if the game reaches a position of 0, Çağatay wins, but if the game continues indefinitely without ever reaching 0, Batu wins.
Let's explain the modulo d arithmetic in the game:
1- When Batu makes a move, he adds 1,2, or 3 to the current position. The resulted number is then computed modulo d. This ensures that the new position remains withing the field {0, 1, 2, ..., d-2, d-1). For example, if d=9 and the current position is 7. If Batu decides to add 3, the new position would be (7 + 3) mod 9 = 10 mod 9 = 1.
2- When Çağatay makes a move, he multiplies the current position by 2, 3, 5, or 7. The resulting number is then computed modulo d. This also ensures that the new position remains within the field (0, 1, 2, ..., d-2, d-1). Let's continue from the position 1, if Boris decides to multiply by 5, the new position would be (1 * 5) mod 9 = 5 mod 9 = 5.
Let's consider d=9 and see if there is a winning strategy for a player. Batu's goal is to prevent Çağatay from reaching 0. Since Çağatay can only reach 0 by multiplying a multiple of 3 by 3, Batu's winning strategy involves always moving to a number that is not a multiple of 3. By doing so, he ensures that Çağatay never gets the opportunity to multiply 3 by 3 and reach 0. Batu can always find a move that results in a number that is not a multiple of 3, as he can add 1, 2, or 3 to the current position. As long as Batu follows this strategy, Çağatay will never be able to reach 0.
Now for the values of d equal to 4, 6, 7, 17, and 30, identify which player has a strategy that guarantees a win.

Answer 1

Batu has a winning strategy for d=7 and d=17, as they are prime and not multiples of 3, allowing him to avoid positions that are multiples of 3. Çağatay has a winning strategy for d=4, d=6, and d=30, because, with these values, Batu cannot prevent Çağatay from eventually reaching 0 by careful multiplications.

Step-by-step explanation:

To determine which player has a winning strategy for the given values of d (4, 6, 7, 17, and 30), we need to analyze the moves each player can make and their effects on the game's outcome.

Batu's moves add to the current position (1, 2, or 3), and Çağatay's moves multiply the current position by one of (2, 3, 5, or 7), all performed modulo d. For Çağatay to win, the game must reach the position 0, which implies Batu must avoid making a move that results in a multiple of 3, because only then can Çağatay multiply it by 3 to reach 0 modulo d.

Considering the values of d:

• For d=4 and d=6, both are multiples of 2 and 3, and Batu cannot prevent Çağatay from reaching 0 because any move Batu makes can be countered by either a multiplication by 2 or 3 to eventually reach 0.
• For d=7 and d=17, both are prime numbers and not multiples of 2 or 3. Batu can prevent the game from ever reaching 0 by adding a number that results in a position that is not a multiple of 3.
• For d=30, it's a multiple of 3, and there exists a number that Çağatay can use to multiply such that eventually, the game reaches 0, giving Çağatay a winning strategy.