Help finding winning strategies in the following tile game?

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In summary, the game involves Player 1 and Player 2 taking turns placing a piece on a board with n tiles. Player 1 always goes first and issues an order, which is a number between 0 and n-1, while Player 2 chooses a tile and moves the piece either left or right according to the order. Player 2 cannot change direction mid-move. Player 2 wins if he can follow all orders for d turns, while Player 1 wins if he can issue an order that Player 2 cannot follow. For d=1 or d=2, there are no winning strategies for Player 1, but for d>=3, there are. The conditions for these "bad triples" depend on n,
  • #1
jjuren
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I'm hoping someone can help me figure out how to describe all winning strategies for "Player 1" in the following game:

Consider a board with $n$ tiles arranged in a row. Player 1 and Player 2 each have $d$ turns, and Player 1 always goes first. On the first turn, Player 1 "issues an order," or in other words, gives a number between $0$ and $n-1,$ and Player 2 chooses a tile on the board to place his piece and then moves either left or right according to the order given by Player 1. Player 2 is never allowed to change direction mid-move. In each subsequent turn, Player 1 issues another order (again, always a number between $0$ and $n-1$), and Player 2 must attempt to execute the order by moving left or right from the square he finished on during the previous turn. Player 2 wins the game if he is able to follow each order and last until the $d$ turns are over, while Player 1 wins if he is able to issue an order that Player 2 cannot follow.

Certainly, if $d=1$ or $d=2$ there are no winning strategies for Player 1, while for $d \geq 3$ there are. So my question is, what are they? Or, put it another way, how can we be sure that a sequence of $d$ orders issued by Player 1 will not be able to be followed by Player 2?

It’s not too difficult to come up with particular winning strategies for Player 1, for example if the sequences $n-1, 1, n-1$ or $n-1, 2, n-1$ appeared among the $d$ orders given by Player 1, that would give a winning strategy. More generally a “high-med-high” strategy is always going to do it – Player 2 would, if acting optimally, place his piece on the edges to begin (to allow for the greatest number of possible moves), so a strategy which eventually forces him closer to the middle of the board and then follows up with a high number Player 2 would be unable to comply. The question I’m really asking then is what are the conditions, in terms of $n,$ which characterize these “bad triples.” How low can the first and last numbers in the triple be relative the middle one?

Thanks!

P.S. This game is related to another problem I was interested in solving:

Which sequences of $d$ integers $e_1, e_2, \dots, e_d,$ with $0 \leq e_i \leq n-1,$ create a system of equations $e_i = |x_i - x_{i+1}|,$ $i=1,2, \dots, n,$ which admits at least one solution in $x_1, x_2, x_3, \dots, x_{d+1},$ with $0 \leq x_i \leq n-1?$ The losing strategies for Player 1 in the above game would yield such sequences, for as Player 2 executes the "orders" given by Player 1 (which would form a sequence $\{e_i\}_{i=1}^d$), he generates a solution (in $x_j$) to the above system of equations.
 
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I do understand the game from your description (at least 90%). However, I don't quite understand the question you are asking so I am just sharing my view on the game here.

Let's say after first turn, the piece is not at the edge of the board, Player 1 will certainly win by issuing an order of n-1. Thus, the only way for Player 2 to survive 2nd turn is to move the piece to the edge of the board during 1st turn.

With the above understanding, it is easy to see that Player 2, upon hearing the number given by player1, will always place the piece somewhere in the board such that the piece can be moved to the edge of the board after the order is executed.

In the beginning of the 2nd turn, the piece is at the edge. Now, Player 1 can issue an order of any number, say r. Player 2 will then move the piece accordingly. During the 3rd turn, Player 1 can then issue an order of n-r to win because the total number of movements of the last 2 turns is r + (n-r) = n, which is greater than n - 1.
 
  • #3
imiuru said:
I do understand the game from your description (at least 90%). However, I don't quite understand the question you are asking so I am just sharing my view on the game here.

Let's say after first turn, the piece is not at the edge of the board, Player 1 will certainly win by issuing an order of n-1. Thus, the only way for Player 2 to survive 2nd turn is to move the piece to the edge of the board during 1st turn.

With the above understanding, it is easy to see that Player 2, upon hearing the number given by player1, will always place the piece somewhere in the board such that the piece can be moved to the edge of the board after the order is executed.

In the beginning of the 2nd turn, the piece is at the edge. Now, Player 1 can issue an order of any number, say r. Player 2 will then move the piece accordingly. During the 3rd turn, Player 1 can then issue an order of n-r to win because the total number of movements of the last 2 turns is r + (n-r) = n, which is greater than n - 1.

Putting that together, player 1's supposed winning strategy would be ( n-1, r, n-r )
But if r > n/2 then player 2's compliant implementation could be ( 1, n, n-r, 2n-2r )

I think that jjuren has already worked this much out. He is after constraints like "r > n/2" but with considerably more generality.
 

1. How do I determine the best starting move in the tile game?

The best starting move in the tile game depends on various factors such as the position of your opponent's tiles, the distribution of tiles on the board, and your own personal strategy. It is important to carefully analyze the board and consider all possible future moves before making your first move.

2. What is the importance of controlling the center tiles in the tile game?

Controlling the center tiles in the tile game gives you a strategic advantage as it allows you to have a better control over the board and limit your opponent's options for future moves. It also increases your chances of obtaining high scoring combinations.

3. How can I prevent my opponent from obtaining high scoring combinations in the tile game?

One way to prevent your opponent from obtaining high scoring combinations is to block their access to certain areas of the board. This can be done by strategically placing your own tiles in key positions. Additionally, try to anticipate your opponent's moves and block their potential scoring opportunities.

4. Is it better to prioritize obtaining high scoring combinations or blocking my opponent's moves in the tile game?

This ultimately depends on your personal strategy and the current state of the game. If you have a strong lead, it may be more beneficial to focus on blocking your opponent's moves to maintain your advantage. However, if you are behind, it may be more beneficial to prioritize obtaining high scoring combinations to catch up.

5. Are there any common patterns or strategies to look out for in the tile game?

Yes, there are some common patterns and strategies that can be used in the tile game. For example, creating a diagonal line of tiles can increase your chances of obtaining high scoring combinations. Additionally, try to keep your tiles connected to each other to have more options for future moves.

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