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? | ||
+ | |||
+ | \[ \begin{array}[b]{|c|c|c|c|c|c} | ||
+ | \vdots &\vdots&\vdots&\vdots&\vdots&\iddots\\\hline | ||
+ | Q&R&P&Q&R&\cdots\\\hline | ||
+ | P&Q&R&P&Q&\cdots\\\hline | ||
+ | R&P&Q&R&P&\cdots\\\hline | ||
+ | Q&R&P&Q&R&\cdots\\\hline | ||
+ | P&Q&R&P&Q&\cdots\\\hline | ||
+ | \end{array} \] | ||
+ | |||
+ | <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{C}}~133</math> Ps, <math>134</math> Qs, <math>133</math> Rs | ||
+ | <math>\boxed{\text{D}}~134</math> Ps, <math>132</math> Qs, <math>134</math> Rs | ||
+ | <math>\boxed{\text{E}}~134</math> Ps, <math>133</math> Qs, <math>133</math> Rs | ||
+ | |||
== Solution 1 == | == Solution 1 == | ||
Revision as of 21:22, 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?
\[ \begin{array}[b]{|c|c|c|c|c|c} \vdots &\vdots&\vdots&\vdots&\vdots&\iddots\\\hline Q&R&P&Q&R&\cdots\\\hline P&Q&R&P&Q&\cdots\\\hline R&P&Q&R&P&\cdots\\\hline Q&R&P&Q&R&\cdots\\\hline P&Q&R&P&Q&\cdots\\\hline \end{array} \]
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)