2020 AMC 8 Problems/Problem 4
- 1 Problem
- 2 Solution 1 (Pattern of the Rows)
- 3 Solution 2 (Pattern of the Bands)
- 4 Solution 3 (Pattern of the Bands)
- 5 Solution 4 (Brute Force)
- 6 Video Solution by WhyMath
- 7 Video Solution by The Learning Royal
- 8 Video Solution by Interstigation
- 9 Video Solution by North America Math Contest Go Go Go
- 10 See also
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 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
Solution 3 (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.). We therefore predict that that the fourth hexagon has dots.
For positive integers let denote the number of dots in the th hexagon. We have and
It follows that and
Solution 4 (Brute Force)
From the full diagram below, the answer is ~MRENTHUSIASM
Video Solution by WhyMath
Video Solution by The Learning Royal
~The Learning Royal
Video Solution by Interstigation
Video Solution by North America Math Contest Go Go Go
~North America Math Contest Go Go Go
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