A game of 11 marbles wherein each player can either pick one or two marbles from the total. Starting from Player A & then Player B alternatively. This continues till all the marbles are picked. The winner is the one having odd number of marbles.
What is the strategy to be followed for Player A & B to win?. What happens for higher total number of marbles (13, 15 etc )?
My guess is that player A wins if the number of marbles is of the form 4n+1 and player B if it is of the form 4n+3. Generally whether you can win in a particular situation depends on the number of marbles you own modulo 2 and the number of marbles left on the table modulo 4. Here are the winning positions AE plus winning moves. First number is number of marbles owned mod 2, second is number of marbles left mod 4:
A: 0/1: take 1 : 1/0
B: 0/2: take 1 : 1/1
C: 1/0: take 1: 0/3
D: 1/2: take 2: 1/0
E: 1/3: take 2: 1/1
Whatever the next player does, we will end playing another one of these winning positions in the next move:
A+2 moves: 1/3 or 1/2
B+2 moves: 1/0 or 1/3
C+2 moves: 0/2 or 0/1
D+2 moves: 1/3 or 1/2
E+2 moves: 1/0 or 1/3
Of course in situation C it is possible that no marbles are left, in which case we have won.
I should add that this could be rubbish, cause I am still suffering a little from my
cork experiment, so be gentle with your comments...

Posted by JLo
on 20061003 17:27:01 