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

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==Video Solution by OmegaLearn (Using Cyclic Patterns)==
 
==Video Solution by OmegaLearn (Using Cyclic Patterns)==
 
https://youtu.be/83FnFhe4QgQ
 
https://youtu.be/83FnFhe4QgQ
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==Video Solution by Magic Square==
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https://youtu.be/-N46BeEKaCQ?t=3990
  
 
==See Also==  
 
==See Also==  
 
{{AMC8 box|year=2023|num-b=15|num-a=17}}
 
{{AMC8 box|year=2023|num-b=15|num-a=17}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 10:37, 25 January 2023

Problem

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?

\[%%Table made by Technodoggo \begin{array}[b]{|c|c|c|c|c|c} \vdots &\vdots&\vdots&\vdots&\vdots&\iddots\\\hline Q&R&P&Q&R&\cdots\\\hline P&Q&R&P&Q&\cdots\\\hline R&P&Q&R&P&\cdots\\\hline Q&R&P&Q&R&\cdots\\\hline P&Q&R&P&Q&\cdots\\\hline \end{array}\]


$\textbf{(A)}~132\text{ Ps, }134\text{ Qs, }134\text{ Rs}$

$\textbf{(B)}~133\text{ Ps, }133\text{ Qs, }134\text{ Rs}$

$\textbf{(C)}~133\text{ Ps, }134\text{ Qs, }133\text{ Rs}$

$\textbf{(D)}~134\text{ Ps, }132\text{ Qs, }134\text{ Rs}$

$\textbf{(E)}~134\text{ Ps, }133\text{ Qs, }133\text{ Rs}$

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

Animated Video Solution

https://youtu.be/1tnMR0lNEFY

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

Video Solution by OmegaLearn (Using Cyclic Patterns)

https://youtu.be/83FnFhe4QgQ

Video Solution by Magic Square

https://youtu.be/-N46BeEKaCQ?t=3990

See Also

2023 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AJHSME/AMC 8 Problems and Solutions

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