Difference between revisions of "2023 AMC 8 Problems/Problem 16"
m (tried to fix latex) |
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| 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}</math> |
'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{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{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> |
== Solution 1 == | == Solution 1 == | ||
| − | In our < | + | 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> |
Revision as of 21:35, 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?
![$\begin{array}[b]{|c|c|c|c|c|c} \vdots &\vdots&\vdots&\vdots&\vdots&\iddots\\\hline Q&R&P&Q&R&\cdots\\\hline P&Q&R&P&Q&\cdots\\\hline R&P&Q&R&P&\cdots\\\hline Q&R&P&Q&R&\cdots\\\hline P&Q&R&P&Q&\cdots\\\hline \end{array}$](http://latex.artofproblemsolving.com/4/3/5/435420b40a00c0721a0b744419b425ea9264fdd0.png)
'Table made by Technodoggo'
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)
