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 )?
The table comes out:
1 4 8 1 0 8 5
1 6 6 1 0 6 7
1 8 4 1 0 4 9
2 3 8 2 0 8 5
2 4 7 1 1 7 5 0 5 8
2 5 6 2 0 6 7
2 6 5 1 1 5 7 0 7 6
2 7 4 2 0 4 9
3 3 7 2 1 7 5 0 5 8
3 5 5 2 1 5 7 0 7 6
4 3 6 1 3 6 4 2 4 7 1 4 8
4 5 4 1 3 4 6 2 6 5 1 6 6
5 2 6 1 4 6 3 3 3 7 2 3 8
5 2 6 2 3 6 4 2 4 7 1 4 8
5 4 4 1 4 4 5 3 5 5 2 5 6
5 4 4 2 3 4 6 2 6 5 1 6 6
6 2 5 1 5 5 3 4 3 6 3 3 7
6 3 4 2 4 4 5 3 5 5 2 5 6
6 4 3 1 5 3 5 4 5 4 3 5 5
7 3 3 2 5 3 5 4 5 4 3 5 5
8 1 4 1 7 4 2 6 2 5 5 2 6
8 3 2 1 7 2 4 6 4 3 5 4 4
9 2 2 1 8 2 3 7 3 3 6 3 4
9 2 2 2 7 2 4 6 4 3 5 4 4
10 1 2 2 8 2 3 7 3 3 6 3 4
10 2 1 1 9 1 3 8 3 2 7 3 3
11 1 1 2 9 1 3 8 3 2 7 3 3
13 0 0 1 12 0 1 11 1 1 10 1 2
13 0 0 2 11 0 2 10 2 1 9 2 2
So the first player has the win, initially taking either 1 or 2 marbles.
It looks as if, if the first player takes 2 initially, he can then mirror the opponent's strategy (after all, it's just like the 11-marble game as the second player). But if the first player decides to take 1 initially, it looks like it would be an anti-mirror strategy, having the two players' moves add to 3.
For 15 marbles the second player has a win again, mirroring his opponent:
1 4 10 1 0 10 5
1 6 8 1 0 8 7
1 8 6 1 0 6 9
2 4 9 1 1 9 5 0 5 10
2 5 8 2 0 8 7
2 6 7 1 1 7 7 0 7 8
2 7 6 2 0 6 9
2 8 5 1 1 5 9 0 9 6
3 5 7 2 1 7 7 0 7 8
3 7 5 2 1 5 9 0 9 6
4 3 8 1 3 8 4 2 4 9 1 4 10
4 5 6 1 3 6 6 2 6 7 1 6 8
4 7 4 1 3 4 8 2 8 5 1 8 6
5 4 6 1 4 6 5 3 5 7 2 5 8
5 4 6 2 3 6 6 2 6 7 1 6 8
5 6 4 1 4 4 7 3 7 5 2 7 6
5 6 4 2 3 4 8 2 8 5 1 8 6
6 3 6 2 4 6 5 3 5 7 2 5 8
6 4 5 1 5 5 5 4 5 6 3 5 7
6 5 4 2 4 4 7 3 7 5 2 7 6
6 6 3 1 5 3 7 4 7 4 3 7 5
7 3 5 2 5 5 5 4 5 6 3 5 7
7 5 3 2 5 3 7 4 7 4 3 7 5
8 3 4 1 7 4 4 6 4 5 5 4 6
9 2 4 1 8 4 3 7 3 5 6 3 6
9 2 4 2 7 4 4 6 4 5 5 4 6
9 4 2 1 8 2 5 7 5 3 6 5 4
9 4 2 2 7 2 6 6 6 3 5 6 4
10 1 4 2 8 4 3 7 3 5 6 3 6
10 2 3 1 9 3 3 8 3 4 7 3 5
10 3 2 2 8 2 5 7 5 3 6 5 4
11 1 3 2 9 3 3 8 3 4 7 3 5
12 1 2 1 11 2 2 10 2 3 9 2 4
13 0 2 1 12 2 1 11 1 3 10 1 4
13 0 2 2 11 2 2 10 2 3 9 2 4
14 0 1 1 13 1 1 12 1 2 11 1 3
At various times the player has a choice of two winning moves, so variance from the mirroring strategy is apparently ok at these points in the game.
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Posted by Charlie
on 2006-10-04 09:34:28 |
for 13 marbles, 15 marbles
