Question

Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move: $(3,2),(2,1,2),(4),(4,1),(2,2),$ or $(1,1,2).$

Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?

$(6,1,1)$

$(6,2,1)$

$(6,2,2)$

$(6,3,1)$

$(6,3,2)$

Solution:

For a single wall of length $n$ , compute its Sprague-Grundy value from the possible moves. For the wall lengths needed here, the values are

$g(1)=1,\quad g(2)=2,\quad g(3)=3,\quad g(4)=1,\quad g(5)=4,\quad g(6)=3.$

For several walls, the position is losing for the player to move exactly when the xor of the wall values is $0$ . Evaluating the choices gives

$(6,1,1): 3\oplus1\oplus1=3,$

$(6,2,1): 3\oplus2\oplus1=0,$

$(6,2,2): 3\oplus2\oplus2=3,$

$(6,3,1): 3\oplus3\oplus1=1,$

$(6,3,2): 3\oplus3\oplus2=2.$

Only $(6,2,1)$ is losing for the player to move, so Beth has a guaranteed win exactly for that starting configuration.

Thus, the answer is B .

2021 AMC 10B Spring Problem 24

Solution:

Problem 24 in Other Years