# the game of dividing towers

If we have K number of towers at different heights. Two players play against each other. Each player in turn choose one tower that have more than 1 cube, and divides it to two separate towers (not necessarily equal). The winner player is the player who's after his turn all the towers in the game are at a height of 1 cube. What strategy for winning the game could be developed?

Rephrase the game. Say you have a single tall tower (or a long row) of blocks, and a move consists of a player putting a divider between two blocks where there wasn't already a divider. Starting my game without any dividers corresponds to starting your game with a single tower, and starting your game with $$K$$ towers corresponds to starting my game with $$K-1$$ dividers already inserted.
Alternately, take your game as it is phrased and look at the number of blocks that have at least one other block on top of them. No matter what move a player makes, tis number will decrease by $$1$$, and the game ends when it reaches $$0$$. Again we see that there isn't really any strategy here, and the winning player is completely based on whether this number is even or odd.