It is obvious that for the first player (A) to win he must remove, in the first step, less than 6 pebbles (as
he is not allowed to remove all the 12 pebbles).
So the number of pebbles A can remove in order to win will be 2,3,4 ,5.
Removing 1 pebble by A, in the first step, will ensure that B is the winner.
A will win, after B removes his pebbles, each time the number of the remaining pebbles will be 5 or 6.
From such a position A always win as shown below. The game is “don’t be greedier”
left A left B left A left B left A left B left A left A
5
1 4
1 3 1
2 1 1
6
2 4
2 2 2 0
6
2 4
1 3 1
2 1
1 0
A WINS
A WINS
1
1 0
A WINS
The position of 5 or 6 remaining pebbles can occur at any step during the play.