Difference between revisions of "2023 AMC 8 Problems/Problem 16"
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Notice that rows <math>x</math> and <math>x+3</math> are the same, for any <math>1 \leq x \leq 17</math>. Additionally, rows <math>1</math>, <math>2</math>, and <math>3</math> collectively contain the same number of <math>P</math>s, <math>Q</math>s, and <math>R</math>s, because the letters are just substituted for one another. Therefore, the number of <math>P</math>s, <math>Q</math>s, and <math>R</math>s in the first <math>18</math> rows is <math>120</math>. The first row has <math>7P</math>, <math>7Q</math>, and <math>6R</math>, and the second row has <math>6P</math>, <math>7Q</math>, and <math>7R</math>. Adding these up, we obtain <math>\boxed{\textbf{(C)}~133\text{ Ps, }134\text{ Qs, }133\text{ Rs}}</math>. | Notice that rows <math>x</math> and <math>x+3</math> are the same, for any <math>1 \leq x \leq 17</math>. Additionally, rows <math>1</math>, <math>2</math>, and <math>3</math> collectively contain the same number of <math>P</math>s, <math>Q</math>s, and <math>R</math>s, because the letters are just substituted for one another. Therefore, the number of <math>P</math>s, <math>Q</math>s, and <math>R</math>s in the first <math>18</math> rows is <math>120</math>. The first row has <math>7P</math>, <math>7Q</math>, and <math>6R</math>, and the second row has <math>6P</math>, <math>7Q</math>, and <math>7R</math>. Adding these up, we obtain <math>\boxed{\textbf{(C)}~133\text{ Ps, }134\text{ Qs, }133\text{ Rs}}</math>. | ||
~mathboy100 | ~mathboy100 | ||
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==Animated Video Solution== | ==Animated Video Solution== | ||
Revision as of 14:51, 26 January 2023
Contents
Problem
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?
![\[%%Table made by Technodoggo \begin{array}[b]{|c|c|c|c|c|c} \vdots &\vdots&\vdots&\vdots&\vdots&\iddots\\\hline Q&R&P&Q&R&\cdots\\\hline P&Q&R&P&Q&\cdots\\\hline R&P&Q&R&P&\cdots\\\hline Q&R&P&Q&R&\cdots\\\hline P&Q&R&P&Q&\cdots\\\hline \end{array}\]](http://latex.artofproblemsolving.com/f/2/d/f2d8e40f2930dd5cf94e2706f4c428c97c226960.png)
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
Solution 2
We think about which letter is in the diagonal with 
of a letter. We find that it is 
. The rest of the grid with the P's and R's is symmetrical, so therefore, the answer is .
Solution by ILoveMath31415926535
Solution 3
Notice that rows and 
are the same, for any 
. Additionally, rows 
, 
, and 
collectively contain the same number of 
s, 
s, and 
s, because the letters are just substituted for one another. Therefore, the number of s, s, and s in the first 
rows is 
. The first row has 
, 
, and 
, and the second row has 
, , and 
. Adding these up, we obtain .
~mathboy100
Animated Video Solution
~Star League (https://starleague.us)
Video Solution by OmegaLearn (Using Cyclic Patterns)
Video Solution by Magic Square
https://youtu.be/-N46BeEKaCQ?t=3990
See Also
| 2023 AMC 8 (Problems • Answer Key • Resources) | ||
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
