GPT4 fails even in extremely simple games. ”There are four pebbles on the 2 x 2 grid. Each turn, one player can remove one pebble, or two nearby pebbles. If these were the last pebbles, he wins. If two players play, who has the winning strategy?”
🤖”In this game, the first player has a winning strategy. The strategy is to always leave an odd number of pebbles for the opponent to take on their turn. This can be achieved by removing one pebble on the first turn, and then mirroring the opponent’s moves throughout the game. Eventually, the first player will be left with one pebble, which they can remove to win the game. “
I guess it just have failed to understand the rules.
With some helpful questions and commands (eg. “which was the last stone a given player took to win?” and “write out the full tree of every possible move of the game!”) it is able to solve it:
[..]
🧔‍♂️What if the players play to win?
🤖If both players play optimally to win, the game will unfold as follows:
Player 1 takes a stone from any cell (A, B, C, or D): 1.1. Player 1 does NOT take a neighboring stone (3 stones left):
sql Copy code 1.1.1. Player 2 takes the stone with two neighbors:
1.1.1.1. Player 2 does NOT take an additional stone (2 non-neighboring stones left): - Player 1 takes one stone (1 stone left). - Player 2 takes the last stone. Player 2 wins.
* Player 2 will choose this optimal move to win.
1.1.2. Player 2 takes a stone with one neighbor:
1.1.2.1. Player 2 does NOT take an additional stone (2 neighboring stones left): - Player 1 takes one stone and its neighbor (0 stones left). Player 1 wins.
1.1.2.2. Player 2 takes the neighboring stone (1 stone left): - Player 1 takes the last stone. Player 1 wins.
* Player 2 will avoid this option since it doesn’t lead to a win. Player 1 should avoid this option since it leads to a win for Player 2 if both players play optimally. 1.2. Player 1 takes a neighboring stone (2 stones left):
sql Copy code 1.2.1. Player 2 takes one stone:
1.2.1.1. Player 2 does NOT take an additional stone (1 stone left): - Player 1 takes the last stone. Player 1 wins.
1.2.1.2. Player 2 takes the neighboring stone (0 stones left): - Player 2 wins.
* Player 2 will choose this optimal move to win. Player 1 should choose this option because it forces Player 2 to react optimally. It doesn’t guarantee a win, but it’s the best option available. When both players play optimally to win, Player 2 has a winning strategy. Player 1 cannot guarantee a win, and the optimal moves for both players will result in a win for Player 2.
In this 2 x 2 grid pebble game, the player with the winning strategy is the one who goes first, given that they follow the optimal strategy.
Let’s denote the players as Player 1 and Player 2.
Here’s the optimal strategy for Player 1:
In the first turn, Player 1 should remove only one pebble.
Now there are three pebbles left on the grid. Player 2 has two choices:
a. Remove one pebble, leaving two pebbles behind.
b. Remove two nearby pebbles, leaving one pebble behind.
Regardless of Player 2′s choice, Player 1 can now remove the remaining pebbles in their next turn and win the game.
By following this strategy, Player 1 will always win the game, regardless of what moves Player 2 makes
Nope, that’s the wrong solution. Second player wins by mirroring moves. Answer to removing one pebble is removing a pebble diagonally to it, leaving two disconnected pebbles.
GPT4 fails even in extremely simple games.
”There are four pebbles on the 2 x 2 grid. Each turn, one player can remove one pebble, or two nearby pebbles. If these were the last pebbles, he wins. If two players play, who has the winning strategy?”
🤖”In this game, the first player has a winning strategy. The strategy is to always leave an odd number of pebbles for the opponent to take on their turn. This can be achieved by removing one pebble on the first turn, and then mirroring the opponent’s moves throughout the game. Eventually, the first player will be left with one pebble, which they can remove to win the game. “
I guess it just have failed to understand the rules.
With some helpful questions and commands (eg. “which was the last stone a given player took to win?” and “write out the full tree of every possible move of the game!”) it is able to solve it:
Solves it fine now:
In this 2 x 2 grid pebble game, the player with the winning strategy is the one who goes first, given that they follow the optimal strategy.
Let’s denote the players as Player 1 and Player 2.
Here’s the optimal strategy for Player 1:
In the first turn, Player 1 should remove only one pebble. Now there are three pebbles left on the grid. Player 2 has two choices: a. Remove one pebble, leaving two pebbles behind. b. Remove two nearby pebbles, leaving one pebble behind. Regardless of Player 2′s choice, Player 1 can now remove the remaining pebbles in their next turn and win the game. By following this strategy, Player 1 will always win the game, regardless of what moves Player 2 makes
Nope, that’s the wrong solution. Second player wins by mirroring moves. Answer to removing one pebble is removing a pebble diagonally to it, leaving two disconnected pebbles.