Difference between revisions of "2023 AMC 8 Problems/Problem 16"
(→Solution 1) |
(→Solution 2) |
||
| Line 38: | Line 38: | ||
== Solution 2 == | == Solution 2 == | ||
| − | We think about which letter is in the diagonal with 20. We find that it is <math>2(2 + 5 + 8 + 11 + 14 + 17) + 20</math> | + | We think about which letter is in the diagonal with <math>20</math> of a letter. We find that it is <math>2(2 + 5 + 8 + 11 + 14 + 17) + 20 = 134</math>. The rest of the grid with the P's and R's is symmetrical, so therefore, the answer is <math>\boxed{\text{(C)}\hspace{0.1 in} 133, 134, 133}</math> |
Solution by [[User:ILoveMath31415926535|ILoveMath31415926535]] | Solution by [[User:ILoveMath31415926535|ILoveMath31415926535]] | ||
Revision as of 12:32, 25 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
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. 
