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Difference between revisions of "2020 AMC 8 Problems/Problem 4"

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Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?
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<asy>
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size(250);
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real side1 = 1.5;
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real side2 = 4.0;
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real side3 = 6.5;
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real pos = 2.5;
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pair s1 = (-10,-2.19);
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pair s2 = (15,2.19);
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pen grey1 = rgb(100/256, 100/256, 100/256);
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pen grey2 = rgb(183/256, 183/256, 183/256);
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fill(circle(origin + s1, 1), grey1);
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for (int i = 0; i < 6; ++i) {
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draw(side1*dir(60*i)+s1--side1*dir(60*i-60)+s1,linewidth(1.25));
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}
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fill(circle(origin, 1), grey1);
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for (int i = 0; i < 6; ++i) {
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fill(circle(pos*dir(60*i),1), grey2);
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draw(side2*dir(60*i)--side2*dir(60*i-60),linewidth(1.25));
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}
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fill(circle(origin+s2, 1), grey1);
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for (int i = 0; i < 6; ++i) {
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fill(circle(pos*dir(60*i)+s2,1), grey2);
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fill(circle(2*pos*dir(60*i)+s2,1), grey1);
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fill(circle(sqrt(3)*pos*dir(60*i+30)+s2,1), grey1);
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draw(side3*dir(60*i)+s2--side3*dir(60*i-60)+s2,linewidth(1.25));
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}
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</asy>
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Diagram by sircalcsalot
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<math>\textbf{(A) }35 \qquad \textbf{(B) }37 \qquad \textbf{(C) }39 \qquad \textbf{(D) }43 \qquad \textbf{(E) }49</math>

Revision as of 22:28, 17 November 2020

Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon? [asy] size(250); real side1 = 1.5; real side2 = 4.0; real side3 = 6.5; real pos = 2.5; pair s1 = (-10,-2.19); pair s2 = (15,2.19); pen grey1 = rgb(100/256, 100/256, 100/256); pen grey2 = rgb(183/256, 183/256, 183/256); fill(circle(origin + s1, 1), grey1); for (int i = 0; i < 6; ++i) { draw(side1*dir(60*i)+s1--side1*dir(60*i-60)+s1,linewidth(1.25)); } fill(circle(origin, 1), grey1); for (int i = 0; i < 6; ++i) { fill(circle(pos*dir(60*i),1), grey2); draw(side2*dir(60*i)--side2*dir(60*i-60),linewidth(1.25)); } fill(circle(origin+s2, 1), grey1); for (int i = 0; i < 6; ++i) { fill(circle(pos*dir(60*i)+s2,1), grey2); fill(circle(2*pos*dir(60*i)+s2,1), grey1); fill(circle(sqrt(3)*pos*dir(60*i+30)+s2,1), grey1); draw(side3*dir(60*i)+s2--side3*dir(60*i-60)+s2,linewidth(1.25)); } [/asy]

Diagram by sircalcsalot

$\textbf{(A) }35 \qquad \textbf{(B) }37 \qquad \textbf{(C) }39 \qquad \textbf{(D) }43 \qquad \textbf{(E) }49$

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