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Difference between revisions of "2023 AMC 8 Problems/Problem 16"
Line 1: Line 1:
 
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?  
 
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?  
  
−
\begin{array}[b]{|c|c|c|c|c|c}
+
<math>\begin{array}[b]{|c|c|c|c|c|c}
 
\vdots &\vdots&\vdots&\vdots&\vdots&\iddots\\\hline
 
\vdots &\vdots&\vdots&\vdots&\vdots&\iddots\\\hline
 
Q&R&P&Q&R&\cdots\\\hline
 
Q&R&P&Q&R&\cdots\\\hline
Line 10: Line 10:
 
\end{array}
 
\end{array}
 
'Table made by Technodoggo'
 
'Table made by Technodoggo'
−
<math>\boxed{\text{A}}~132\text{ Ps, }134\text{ Qs, }134\text{ Rs}</math>
+
</math>\boxed{\text{A}}~132\text{ Ps, }134\text{ Qs, }134\text{ Rs}<math>
−
<math>\boxed{\text{B}}~133\text{ Ps, }133\text{ Qs, }134\text{ Rs}</math>
+
</math>\boxed{\text{B}}~133\text{ Ps, }133\text{ Qs, }134\text{ Rs}<math>
−
<math>\boxed{\text{C}}~133\text{ Ps, }134\text{ Qs, }133\text{ Rs}</math>
+
</math>\boxed{\text{C}}~133\text{ Ps, }134\text{ Qs, }133\text{ Rs}<math>
−
<math>\boxed{\text{D}}~134\text{ Ps, }132\text{ Qs, }134\text{ Rs}</math>
+
</math>\boxed{\text{D}}~134\text{ Ps, }132\text{ Qs, }134\text{ Rs}<math>
−
<math>\boxed{\text{E}}~134\text{ Ps, }133\text{ Qs, }133\text{ Rs}</math>
+
</math>\boxed{\text{E}}~134\text{ Ps, }133\text{ Qs, }133\text{ Rs}<math>
  
 
== Solution 1 ==
 
== Solution 1 ==
  
−
In our <math>5 \times 5</math> grid we can see there are <math>8</math>, <math>9</math> and <math>8</math> of the letters P, Q and R’s respectively. We can see our pattern between each is <math>x</math>, <math>x+1</math>, <math>x</math> 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 <math>\boxed{\text{(C)}\hspace{0.1 in} 133, 134, 133}</math>  
+
In our </math>5 \times 5<math> grid we can see there are </math>8<math>, </math>9<math> and </math>8<math> of the letters P, Q and R’s respectively. We can see our pattern between each is </math>x<math>, </math>x+1<math>, </math>x<math> 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 </math>\boxed{\text{(C)}\hspace{0.1 in} 133, 134, 133}$  
  
  

Revision as of 22:35, 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?

$\begin{array}[b]{|c|c|c|c|c|c} \vdots &\vdots&\vdots&\vdots&\vdots&\iddots\\\hline Q&R&P&Q&R&\cdots\\\hline P&Q&R&P&Q&\cdots\\\hline R&P&Q&R&P&\cdots\\\hline Q&R&P&Q&R&\cdots\\\hline P&Q&R&P&Q&\cdots\\\hline \end{array} 'Table made by Technodoggo'$\boxed{\text{A}}~132\text{ Ps, }134\text{ Qs, }134\text{ Rs}$$ (Error compiling LaTeX. Unknown error_msg)\boxed{\text{B}}~133\text{ Ps, }133\text{ Qs, }134\text{ Rs}$$ (Error compiling LaTeX. Unknown error_msg)\boxed{\text{C}}~133\text{ Ps, }134\text{ Qs, }133\text{ Rs}$$ (Error compiling LaTeX. Unknown error_msg)\boxed{\text{D}}~134\text{ Ps, }132\text{ Qs, }134\text{ Rs}$$ (Error compiling LaTeX. Unknown error_msg)\boxed{\text{E}}~134\text{ Ps, }133\text{ Qs, }133\text{ Rs}$== Solution 1 ==

In our$ (Error compiling LaTeX. Unknown error_msg)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

Animated Video Solution

https://youtu.be/1tnMR0lNEFY

~Star League (https://starleague.us)

Video Solution by OmegaLearn (Using Cyclic Patterns)

https://youtu.be/83FnFhe4QgQ

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