Determine winner in stack-merging game

Game: Babylon stack merging (two-player, perfect play)

You are given a two-player turn-based game. There are n stacks of tiles. Each stack has:

• a height (positive integer)
• a top color (one of Y , W , G , B )

Initial state

• There are n stacks, each of height = 1 .
• The i -th stack’s top color is colors[i] .

Legal move

On your turn, you must pick two different stacks A and B and merge the entire stack A onto the top of stack B, but only if at least one condition holds:

1. height(A) == height(B) , or
2. topColor(A) == topColor(B)

After merging:

• height(B) = height(A) + height(B)
• topColor(B) = topColor(A) (because A is placed on top)
• stack A is removed (total number of stacks decreases by 1)

Winning condition

• If it is your turn and there is no legal merge , you lose .
• Equivalently, the player who makes the last legal move wins.

Task

Given the initial colors array, determine whether the first player has a winning strategy assuming optimal play from both players.

Return true if the first player can force a win; otherwise return false.

Constraints (you may assume)

• 1 <= n <= 12
• colors[i] is in { 'Y','W','G','B' }

Example (illustrative)

• Input: colors = ["Y","Y"]
• There is one legal move (same top color). First player moves and wins → output true .