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

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==Problem==
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#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 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>
 
 
 
== 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
 

Latest revision as of 01:00, 13 February 2021

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