2023 AMC 8 Problems/Problem 16
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3
- 5 Solution 4
- 6 Solution 5
- 7 Animated Video Solution
- 8 Video Solution by OmegaLearn (Using Cyclic Patterns)
- 9 Video Solution by Magic Square
- 10 Video Solution by Interstigation
- 11 Video Solution by WhyMath
- 12 Video Solution (CREATIVE THINKING!!!)
- 13 See Also
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
Video Solution by WhyMath
~savannahsolver
Video Solution (CREATIVE THINKING!!!)
~Education, the Study of Everything
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.