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Difference between revisions of "2023 AMC 8 Problems/Problem 16"

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The letters P, Q, and R are entered into a <math>20\times20</math> table according to the pattern shown below. How many Ps, Qs, and Rs will appear in the completed table?
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\[ \begin{array}[b]{|c|c|c|c|c|c}
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\vdots &\vdots&\vdots&\vdots&\vdots&\iddots\\\hline
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Q&R&P&Q&R&\cdots\\\hline
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P&Q&R&P&Q&\cdots\\\hline
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R&P&Q&R&P&\cdots\\\hline
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Q&R&P&Q&R&\cdots\\\hline
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P&Q&R&P&Q&\cdots\\\hline
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\end{array} \]
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<math>\boxed{\text{A}}~132</math> Ps, <math>134</math> Qs, <math>134</math> Rs
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<math>\boxed{\text{B}}~133</math> Ps, <math>133</math> Qs, <math>134</math> Rs
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<math>\boxed{\text{C}}~133</math> Ps, <math>134</math> Qs, <math>133</math> Rs
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<math>\boxed{\text{D}}~134</math> Ps, <math>132</math> Qs, <math>134</math> Rs
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<math>\boxed{\text{E}}~134</math> Ps, <math>133</math> Qs, <math>133</math> Rs
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== Solution 1 ==
 
== Solution 1 ==
  

Revision as of 20:22, 24 January 2023

The letters P, Q, and R are entered into a $20\times20$ table according to the pattern shown below. How many Ps, Qs, and Rs will appear in the completed table?

\[ \begin{array}[b]{|c|c|c|c|c|c} \vdots &\vdots&\vdots&\vdots&\vdots&\iddots\\\hline Q&R&P&Q&R&\cdots\\\hline P&Q&R&P&Q&\cdots\\\hline R&P&Q&R&P&\cdots\\\hline Q&R&P&Q&R&\cdots\\\hline P&Q&R&P&Q&\cdots\\\hline \end{array} \]

$\boxed{\text{A}}~132$ Ps, $134$ Qs, $134$ Rs $\boxed{\text{B}}~133$ Ps, $133$ Qs, $134$ Rs $\boxed{\text{C}}~133$ Ps, $134$ Qs, $133$ Rs $\boxed{\text{D}}~134$ Ps, $132$ Qs, $134$ Rs $\boxed{\text{E}}~134$ Ps, $133$ Qs, $133$ Rs

Solution 1

In our $5 \times 5$ grid we can see there are $8$, $9$ and $8$ of the letters P, Q and R’s respectively. We can see our pattern between each is $x$, $x+1$, $x$ for the P, Q and R’s respectively. This such pattern will follow in our bigger example, so we can see that the only answer choice which satisfies this condition is $\boxed{\text{(C)}\hspace{0.1 in} 133, 134, 133}$


(Note: you could also "cheese" this problem by listing out all of the letters horizontally in a single line and looking at the repeating pattern.)


~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat

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