Difference between revisions of "2023 AMC 8 Problems/Problem 16"
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When looking at a pair of <math>Q</math> diagonals of the same length <math>x,</math> there is a total of <math>2x</math> <math>\text{R}</math>s and <math>\text{P}</math>s next to these <math>2</math> diagonals. | When looking at a pair of <math>Q</math> diagonals of the same length <math>x,</math> there is a total of <math>2x</math> <math>\text{R}</math>s and <math>\text{P}</math>s next to these <math>2</math> diagonals. | ||
− | The main diagonal of <math>20</math> <math>\text{Q}</math>s has <math>19</math> <math>\text{P}</math>s and <math>19</math> <math>\text{R}</math>s next to it. Thus the total is <math>x+1</math> <math>\text{Q}</math>s, <math>x</math> <math>\text{P}</math>s, <math>x</math> <math>\text{R}</math>s. Therefore, the answer is <math>\boxed{\textbf{(C)}~133\text{ Ps, }134\text{ Qs, }133\text{ Rs}}.</math> | + | The main diagonal of <math>20</math> <math>\text{Q}</math>s has <math>19</math> <math>\text{P}</math>s and <math>19</math> <math>\text{R}</math>s next to it. Thus, the total is <math>x+1</math> <math>\text{Q}</math>s, <math>x</math> <math>\text{P}</math>s, <math>x</math> <math>\text{R}</math>s. Therefore, the answer is <math>\boxed{\textbf{(C)}~133\text{ Ps, }134\text{ Qs, }133\text{ Rs}}.</math> |
~ERMSCoach | ~ERMSCoach |
Revision as of 02:54, 19 February 2023
Contents
[hide]Problem
The letters and are entered into a table according to the pattern shown below. How many s, s, and s will appear in the completed table?
Solution 1
In our grid we can see there are and of the letters and ’s respectively. We can see our pattern between each is and for the and ’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 's and 's is symmetric. Therefore, the answer is
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 s, s, and s, and the second row has s, s, and s. Adding these up, we obtain .
~mathboy100
Solution 4
From the full diagram below, the answer is This solution is extremely time-consuming and error-prone. Please do not try it in a real competition unless you have no other options.
~MRENTHUSIASM
Solution 5
This solution refers to the full diagram in Solution 4.
Note the diagonals are symmetric. The and diagonals are not symmetric, but are reflections of each other about the diagonals:
- The upper diagonal of length is surrounded by a diagonal of length and an diagonal of length
- The lower diagonal of length is surrounded by a diagonal of length and an diagonal of length
When looking at a pair of diagonals of the same length there is a total of s and s next to these diagonals.
The main diagonal of s has s and s next to it. Thus, the total is s, s, s. Therefore, the answer is
~ERMSCoach
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
Video Solution by Interstigation
https://youtu.be/1bA7fD7Lg54?t=1385
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.