Difference between revisions of "2020 AMC 8 Problems/Problem 4"
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== Video Solution == | == Video Solution == | ||
− | https://www.youtube.com/watch?v= | + | https://www.youtube.com/watch?v=_IjQnXnVKeU |
==Video Solution by WhyMath== | ==Video Solution by WhyMath== |
Revision as of 19:02, 14 October 2021
Contents
Problem
Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?
Solution 1 (Pattern of the Rows)
Looking at the rows of each hexagon, we see that the first hexagon has dot, the second has
dots, and the third has
dots. Given the way the hexagons are constructed, it is clear that this pattern continues. Hence, the fourth hexagon has
dots.
Solution 2 (Pattern of the Bands)
The first hexagon has dot, the second hexagon has
dots, the third hexagon
dots, and so on. The pattern continues since to go from hexagon
to hexagon
we add a new band of dots around the outside of the existing ones, with each side of the band having side length
Thus the number of dots added is
(we subtract
as each of the corner hexagons in the band is counted as part of two sides), confirming the pattern. We therefore predict that that the fourth hexagon has
dots.
Solution 3 (Variant of Solution 2)
The dots in the next hexagon have four bands. From innermost to outermost:
- The first band has
dot.
- The second band has
dots:
dot at each vertex of the hexagon.
- The third band has
dots:
dot at each vertex of the hexagon and
other dot on each edge of the hexagon.
- The fourth band has
dots:
dot at each vertex of the hexagon and
other dots on each edge of the hexagon.
Together, the answer is
~MRENTHUSIASM
Solution 4 (Variant of Solution 2)
Let the number of dots in the first hexagon be By the same argument as in Solution 2, we have
for
Using this, we find that
and
Solution 5 (Brute Force)
From the full diagram below, the answer is
~MRENTHUSIASM
Video Solution
https://www.youtube.com/watch?v=_IjQnXnVKeU
Video Solution by WhyMath
~savannahsolver
Video Solution
Video Solution by Interstigation
https://youtu.be/YnwkBZTv5Fw?t=123
~Interstigation
See also
2020 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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.