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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>.
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| − | <asy>
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| − | /* CREDITS */
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| − | /* Made by samrocksnature */
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| − | /* Modified commas an periods by forester2015 */
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| − | | |
| − | /* Import and Set variables */
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| − | import graph; size(10cm);
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| − | real labelscalefactor = 0.5; /* changes label-to-point distance */
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| − | pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
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| − | pen dotstyle = black; /* point style */
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| − | real xmin = -20.98617651190462, xmax = 71.97119514540032, ymin = -24.802885633158763, ymax = 28.83570218998556; /* image dimensions */
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| − | | |
| − | /* draw figures */
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| − | draw((14,4)--(13.010050506338834,3.0100505063388336), linewidth(1));
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| − | draw((14,4)--(13.010050506338834,4.989949493661166), linewidth(1));
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| − | draw((10,4)--(14,4), linewidth(1));
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| − | draw((4,6)--(8,6), linewidth(1));
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| − | draw((4,2)--(8,2), linewidth(1));
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| − | draw((8,2)--(8,6), linewidth(1));
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| − | draw((4,6)--(4,2), linewidth(1));
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| − | draw((6,6)--(6,2), linewidth(1));
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| − | draw((-6,6)--(-6,2), linewidth(1));
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| − | draw((-6,6)--(2,6), linewidth(1));
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| − | draw((2,6)--(2,2), linewidth(1));
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| − | draw((2,2)--(-6,2), linewidth(1));
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| − | draw((-4,2)--(-4,6), linewidth(1));
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| − | draw((-2,6)--(-2,2), linewidth(1));
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| − | draw((0,2)--(0,6), linewidth(1));
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| − | draw((50,6)--(50,2), linewidth(1));
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| − | draw((50,2)--(58,2), linewidth(1));
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| − | draw((58,2)--(58,6), linewidth(1));
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| − | draw((58,6)--(50,6), linewidth(1));
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| − | draw((52,6)--(52,2), linewidth(1));
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| − | draw((54,6)--(54,2), linewidth(1));
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| − | draw((56,6)--(56,2), linewidth(1));
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| − | draw((32,6)--(32,2), linewidth(1));
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| − | draw((46,2)--(46,6), linewidth(1));
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| − | draw((34,6)--(34,2), linewidth(1));
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| − | draw((36,2)--(36,6), linewidth(1));
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| − | draw((38,6)--(38,2), linewidth(1));
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| − | draw((40,2)--(40,6), linewidth(1));
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| − | draw((42,6)--(42,2), linewidth(1));
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| − | draw((44,2)--(44,6), linewidth(1));
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| − | draw((16,6)--(16,2), linewidth(1));
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| − | draw((28,2)--(28,6), linewidth(1));
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| − | draw((18,6)--(18,2), linewidth(1));
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| − | draw((20,6)--(20,2), linewidth(1));
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| − | draw((22,6)--(22,2), linewidth(1));
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| − | draw((24,6)--(24,2), linewidth(1));
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| − | draw((26,6)--(26,2), linewidth(1));
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| − | draw((16,6)--(22,6), linewidth(1));
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| − | draw((24,6)--(28,6), linewidth(1));
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| − | draw((16,2)--(22,2), linewidth(1));
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| − | draw((24,2)--(28,2), linewidth(1));
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| − | draw((32,6)--(36,6), linewidth(1));
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| − | draw((32,2)--(36,2), linewidth(1));
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| − | draw((38,6)--(40,6), linewidth(1));
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| − | draw((38,2)--(40,2), linewidth(1));
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| − | draw((42,6)--(46,6), linewidth(1));
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| − | draw((42,2)--(46,2), linewidth(1));
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| − | | |
| − | /* dots and labels */
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| − | label(",",(59,2));
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| − | label(".",(60,2));
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| − | label(".",(61,2));
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| − | label(".",(62,2));
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| − | label(",",(29,2));
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| − | label(",",(47,2));
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| − | </asy>
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| − | 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?
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| − | | |
| − | <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>
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| − | | |
| − | == Solution ==
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| − | 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:
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| − | | |
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| − | <math>(2, 2, 2, 1) - (2, 1, 2, 1)</math>
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| − | <math>(1, 3, 2, 1) - (3, 2, 1)</math>
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| − | <math>(4, 2, 1) - (4, 1)</math>
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| − | <math>(6, 1) - (4, 1)</math>
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| − | <math>(5, 2, 1) - (3, 2, 1)</math>
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| − | <math>(4, 1, 2, 1) - (2, 1, 2, 1)</math>
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| − | | |
| − | <math>(3, 2, 2, 1) - (1, 2, 2, 1)</math>
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| − | This proves that <math>(6, 2, 1)</math> is losing for the first player.
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| − | | |
| − | -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.
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| − | | |
| − | == Solution 2 (Process of Elimination)==
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| − | | |
| − | <math>(6,1,1)</math> can be turned into <math>(2,2,1,1)</math> by Arjun, which is symmetric, so Beth will lose.
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| − | <math>(6,3,1)</math> can be turned into <math>(3,1,3,1)</math> by Arjun, which is symmetric, so Beth will lose.
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| − | <math>(6,2,2)</math> can be turned into <math>(2,2,2,2)</math> by Arjun, which is symmetric, so Beth will lose.
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| − | <math>(6,3,2)</math> can be turned into <math>(3,2,3,2)</math> by Arjun, which is symmetric, so Beth will lose.
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| − | That leaves <math>(6,2,1)</math> or <math>\boxed{\textbf{(B)}}</math>.
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| − | | |
| − | == Video Solution by OmegaLearn (Game Theory) ==
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| − | https://youtu.be/zkSBMVAfYLo
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| − | | |
| − | ~ pi_is_3.14
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| − | | |
| − | {{AMC10 box|year=2021|ab=B|num-b=23|num-a=25}}
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