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−  ==Problem==
 +  #redirect [[2021 AMC 12B Problems/Problem 22]] 
−  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 labeltopoint 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>
 
−   
−  == Solution ==
 
−  First we note that symmetrical positions are losing for the player to move. Then we start checking small positions. <math>(n)</math> is always winning for the first player. Furthermore, <math>(3, 2, 1)</math> is losing and so is <math>(4, 1).</math> We look at all the positions created from <math>(6, 2, 1),</math> as <math>(6, 1, 1)</math> is obviously winning by playing <math>(2, 2, 1, 1).</math> There are several different positions that can be played by the first player from <math>(6, 2, 1).</math> They are <math>(2, 2, 2, 1), (1, 3, 2, 1), (4, 2, 1), (6, 1), (5, 2, 1), (4, 1, 2, 1), (3, 2, 2, 1).</math> Now we list refutations for each of these moves:
 
−   
−   
−  <math>(2, 2, 2, 1)  (2, 1, 2, 1)</math>
 
−   
−   
−  <math>(1, 3, 2, 1)  (3, 2, 1)</math>
 
−   
−   
−  <math>(4, 2, 1)  (4, 1)</math>
 
−   
−   
−  <math>(6, 1)  (4, 1)</math>
 
−   
−   
−  <math>(5, 2, 1)  (3, 2, 1)</math>
 
−   
−   
−  <math>(4, 1, 2, 1)  (2, 1, 2, 1)</math>
 
−   
−   
−  <math>(3, 2, 2, 1)  (1, 2, 2, 1)</math>
 
−   
−   
−  This proves that <math>(6, 2, 1)</math> is losing for the first player.
 
−   
−  Note: In general, this game is very complicated. For example <math>(8, 7, 5, 3, 2)</math> is winning for the first player but good luck showing that.
 
−   
−  == Solution 2 (Process of Elimination)==
 
−   
−  == Video Solution by OmegaLearn (Game Theory) ==
 
−  https://youtu.be/zkSBMVAfYLo
 
−   
−  ~ pi_is_3.14
 