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? | 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} | |
\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 8: | Line 8: | ||
Q&R&P&Q&R&\cdots\\\hline | Q&R&P&Q&R&\cdots\\\hline | ||
P&Q&R&P&Q&\cdots\\\hline | P&Q&R&P&Q&\cdots\\\hline | ||
| − | \end{array} | + | \end{array} |
| − | + | 'Table made by Technodoggo' | |
<math>\boxed{\text{A}}~132</math> Ps, <math>134</math> Qs, <math>134</math> Rs | <math>\boxed{\text{A}}~132</math> Ps, <math>134</math> Qs, <math>134</math> Rs | ||
<math>\boxed{\text{B}}~133</math> Ps, <math>133</math> Qs, <math>134</math> Rs | <math>\boxed{\text{B}}~133</math> Ps, <math>133</math> Qs, <math>134</math> Rs | ||
Revision as of 21:24, 24 January 2023
The letters P, Q, and R are entered into a 
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'
Ps, 
Qs, Rs
Ps, 
Qs, Rs
Ps, Qs, Rs
Ps, 
Qs, Rs
Ps, Qs, Rs
Solution 1
In our 
grid we can see there are 
, 
and of the letters P, Q and R’s respectively. We can see our pattern between each is 
, 
, 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 
(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
~Star League (https://starleague.us)
