Suppose two players play a game with two piles of stones. Initially there are five

stones in one pile and seven stones in the other pile. The rules of the game are

that on each turn a player has two choices:

Either $($i$)$ Choose a pile and remove one or more stones from it$.$

$($ii$)$ Remove an equal number of stones from each pile $($at least one from

each$).$

The winner is the person who takes the last stone. Find the Kernel of this game,

and decide which player has winning strategy in this game. NOTE: A player must

remove something on each turn. Removing zero stones or passing on your turn

is NOT an option.