# Difference between revisions of "2021 AMC 10B Problems/Problem 24"

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− | + | ==Problem== | |

− | + | 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 <math>4</math> and <math>2</math> can be changed into any of the following by one move: <math>(3,2),(2, 1, 2),(4),(4, 1),(2, 2),</math> or <math>(1, 1, 2)</math>. | |

− | + | <asy> | |

+ | /* CREDITS */ | ||

+ | /* Made by samrocksnature */ | ||

+ | /* Modified commas an periods by forester2015 */ | ||

+ | |||

+ | /* Import and Set variables */ | ||

+ | import graph; size(10cm); | ||

+ | real labelscalefactor = 0.5; /* changes label-to-point distance */ | ||

+ | pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ | ||

+ | pen dotstyle = black; /* point style */ | ||

+ | real xmin = -20.98617651190462, xmax = 71.97119514540032, ymin = -24.802885633158763, ymax = 28.83570218998556; /* image dimensions */ | ||

+ | |||

+ | /* draw figures */ | ||

+ | draw((14,4)--(13.010050506338834,3.0100505063388336), linewidth(1)); | ||

+ | draw((14,4)--(13.010050506338834,4.989949493661166), linewidth(1)); | ||

+ | draw((10,4)--(14,4), linewidth(1)); | ||

+ | draw((4,6)--(8,6), linewidth(1)); | ||

+ | draw((4,2)--(8,2), linewidth(1)); | ||

+ | draw((8,2)--(8,6), linewidth(1)); | ||

+ | draw((4,6)--(4,2), linewidth(1)); | ||

+ | draw((6,6)--(6,2), linewidth(1)); | ||

+ | draw((-6,6)--(-6,2), linewidth(1)); | ||

+ | draw((-6,6)--(2,6), linewidth(1)); | ||

+ | draw((2,6)--(2,2), linewidth(1)); | ||

+ | draw((2,2)--(-6,2), linewidth(1)); | ||

+ | draw((-4,2)--(-4,6), linewidth(1)); | ||

+ | draw((-2,6)--(-2,2), linewidth(1)); | ||

+ | draw((0,2)--(0,6), linewidth(1)); | ||

+ | draw((50,6)--(50,2), linewidth(1)); | ||

+ | draw((50,2)--(58,2), linewidth(1)); | ||

+ | draw((58,2)--(58,6), linewidth(1)); | ||

+ | draw((58,6)--(50,6), linewidth(1)); | ||

+ | draw((52,6)--(52,2), linewidth(1)); | ||

+ | draw((54,6)--(54,2), linewidth(1)); | ||

+ | draw((56,6)--(56,2), linewidth(1)); | ||

+ | draw((32,6)--(32,2), linewidth(1)); | ||

+ | draw((46,2)--(46,6), linewidth(1)); | ||

+ | draw((34,6)--(34,2), linewidth(1)); | ||

+ | draw((36,2)--(36,6), linewidth(1)); | ||

+ | draw((38,6)--(38,2), linewidth(1)); | ||

+ | draw((40,2)--(40,6), linewidth(1)); | ||

+ | draw((42,6)--(42,2), linewidth(1)); | ||

+ | draw((44,2)--(44,6), linewidth(1)); | ||

+ | draw((16,6)--(16,2), linewidth(1)); | ||

+ | draw((28,2)--(28,6), linewidth(1)); | ||

+ | draw((18,6)--(18,2), linewidth(1)); | ||

+ | draw((20,6)--(20,2), linewidth(1)); | ||

+ | draw((22,6)--(22,2), linewidth(1)); | ||

+ | draw((24,6)--(24,2), linewidth(1)); | ||

+ | draw((26,6)--(26,2), linewidth(1)); | ||

+ | draw((16,6)--(22,6), linewidth(1)); | ||

+ | draw((24,6)--(28,6), linewidth(1)); | ||

+ | draw((16,2)--(22,2), linewidth(1)); | ||

+ | draw((24,2)--(28,2), linewidth(1)); | ||

+ | draw((32,6)--(36,6), linewidth(1)); | ||

+ | draw((32,2)--(36,2), linewidth(1)); | ||

+ | draw((38,6)--(40,6), linewidth(1)); | ||

+ | draw((38,2)--(40,2), linewidth(1)); | ||

+ | draw((42,6)--(46,6), linewidth(1)); | ||

+ | draw((42,2)--(46,2), linewidth(1)); | ||

+ | |||

+ | /* dots and labels */ | ||

+ | label(",",(59,2)); | ||

+ | label(".",(60,2)); | ||

+ | label(".",(61,2)); | ||

+ | label(".",(62,2)); | ||

+ | label(",",(29,2)); | ||

+ | label(",",(47,2)); | ||

+ | </asy> | ||

+ | 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? | ||

+ | |||

+ | <math>\textbf{(A)} ~(6, 1, 1) \qquad\textbf{(B)} ~(6, 2, 1) \qquad\textbf{(C)} ~(6, 2, 2) \qquad\textbf{(D)} ~(6, 3, 1) \qquad\textbf{(E)} ~(6, 3, 2)</math> |

## Revision as of 19:44, 11 February 2021

## Problem

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 and can be changed into any of the following by one move: or . 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?