2023 AMC 8 Problems/Problem 16

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

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

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 $\boxed{\text{(C)}\hspace{0.1 in} 133, 134, 133}$