AMC 8 · 2020 · #4

Easy mode Grade 4

Problem

Picture three hexagons made of dots, getting bigger from left to right. (The first hexagon is a single dot. The next one is a ring of dots around that dot. The third is another ring around that.)

The pattern keeps going the same way. Each new hexagon adds one more ring of dots around the outside of the last one.

How many dots will the next hexagon (the fourth one) have?

$\textbf{(A) }35 \qquad \textbf{(B) }37 \qquad \textbf{(C) }39 \qquad \textbf{(D) }43 \qquad \textbf{(E) }49$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: Three hexagonal dot figures are shown, each one band larger than the last. The 1st hexagon has $1$ dot, the 2nd has $7$ dots, and the 3rd has $19$ dots. If the pattern continues by adding one more ring of dots each time, how many dots are in the 4th (next) hexagon?

Givens: 1st hexagon: $1$ dot (a single center dot); 2nd hexagon: $3$ rows with $2, 3, 2$ dots, totaling $7$; 3rd hexagon: $5$ rows with $3, 4, 5, 4, 3$ dots, totaling $19$; Each successive hexagon adds one more outer band of dots; Answer choices: (A) $35$, (B) $37$, (C) $39$, (D) $43$, (E) $49$

Unknowns: The total number of dots in the 4th hexagon

Understand

Plan

Primary tool: #5 Look for a Pattern

Secondary: #9 Solve an Easier Related Problem, #2 Make a Systematic List, #3 Eliminate Possibilities

The picture only gives us hexagons of size $1, 2, 3$, and asks about size $4$ — a textbook "what comes next?" setup, so Tool #5 (Look for a Pattern) is the spine of the solution. Tool #9 (Easier Related Problem) is already half-done for us: the first three hexagons ARE the easier cases, and our job is to read them carefully and generalize. Tool #2 (Systematic List) keeps the row counts honest — for hexagon $n$, list the rows from top to middle to bottom in order, then add. Tool #3 (Eliminate Possibilities) is the standard final check on a multiple-choice problem — confirm our total matches exactly one of A-E. We deliberately avoid Tool #13 (Algebra) and Tool #14 (Finite Differences) on the totals $1, 7, 19, \ldots$ because the row-by-row structure is visually obvious and a Grade 4 pattern-recognition argument is enough.

Execute — Answer: B

#9 Solve an Easier Related Problem 4.OA.C.5 Step 1

Read the three given hexagons row by row.
Hexagon $1$ has one row of $1$ dot.
Hexagon $2$ has $3$ rows: $2, 3, 2$.
Hexagon $3$ has $5$ rows: $3, 4, 5, 4, 3$.
Notice the row counts are symmetric and the middle row is the longest.

$$1;\quad 2,3,2;\quad 3,4,5,4,3$$

💡 The three pictures we were given are themselves the "easier related problems" — reading them carefully is the Grade 4 "describe a pattern" skill.

#5 Look for a Pattern 4.OA.C.5 Step 2

Find the rule.
In hexagon $n$ the rows go up by $1$ from $n$ to $2n-1$ in the middle, then back down by $1$ to $n$.
So hexagon $n$ has $2n-1$ rows whose dot counts are $n, n+1, \ldots, 2n-1, \ldots, n+1, n$.
Quick check: $n=2$ gives $2, 3, 2$ (correct), and $n=3$ gives $3, 4, 5, 4, 3$ (correct).

$$n=2:\ 2,3,2\ \checkmark\qquad n=3:\ 3,4,5,4,3\ \checkmark$$

💡 Spotting that the next row count is always one more than the previous (until the middle, then one less) is exactly Grade 4 "generate a number pattern following a rule."

#2 Make a Systematic List 4.OA.C.5 Step 3

Apply the rule with $n=4$.
The $4$th hexagon has $2(4)-1 = 7$ rows, and the dot counts are $4, 5, 6, 7, 6, 5, 4$.
List them systematically from top to bottom to make sure none are skipped or repeated.

$$4,\ 5,\ 6,\ 7,\ 6,\ 5,\ 4$$

💡 A systematic top-to-bottom list guarantees we count every row exactly once.

#5 Look for a Pattern 3.NBT.A.2 Step 4

Add the row counts to get the total number of dots.
Pair the symmetric outer rows first to keep the arithmetic friendly: $4+4=8$, $5+5=10$, $6+6=12$, plus the middle row $7$.

$$4+5+6+7+6+5+4 = (4+4)+(5+5)+(6+6)+7 = 8+10+12+7 = 37$$

💡 Adding seven small whole numbers within $1000$ is the Grade 3 fluent addition standard — no decimals, no fractions.

#3 Eliminate Possibilities 4.OA.C.5 Step 5

Match the total to the answer choices.
$37$ is choice (B).

$$37 \;\Rightarrow\; \textbf{(B)}$$

💡 On multiple choice, the final move is always to confirm our number is in the list.

[1] #9 4.OA.C.5 Read the three given hexagons row by row. Hexagon $1$ has one row of $1$ dot. He

[2] #5 4.OA.C.5 Find the rule. In hexagon $n$ the rows go up by $1$ from $n$ to $2n-1$ in the mi

[3] #2 4.OA.C.5 Apply the rule with $n=4$. The $4$th hexagon has $2(4)-1 = 7$ rows, and the dot

[4] #5 3.NBT.A.2 Add the row counts to get the total number of dots. Pair the symmetric outer row

[5] #3 4.OA.C.5 Match the total to the answer choices. $37$ is choice (B).

Review

Reasonableness: The totals so far are $1, 7, 19, 37$. The jumps between them are $6, 12, 18$ — a clean arithmetic progression of $+6$ each time. That makes sense because going from hexagon $n$ to hexagon $n+1$ wraps a new ring around the outside, and a hexagonal ring of "radius" $n$ contains exactly $6n$ extra dots. So hexagon $4$ should be $19 + 6\times 3 = 19 + 18 = 37$ — matching choice (B). The next term would be $37 + 24 = 61$, which is the formula for centered hexagonal numbers $3n^2 - 3n + 1$ evaluated at $n=5$: a satisfying sanity check.

Alternative: Tool #14 (Finite Differences) on the totals $1, 7, 19, ?$ gives first differences $6, 12$, second difference $6$. Assuming constant second differences, the next first difference is $12+6=18$, so the next total is $19+18=37$ — same answer (B) by a different route. This is a nice cross-check but it skips the visual structure that makes the problem easy for a younger student.

CCSS standards used (min grade 4)

4.OA.C.5 Generate a number or shape pattern following a given rule (Reading the row counts of hexagons $1, 2, 3$, conjecturing the rule "rows climb $1$ at a time up to $2n-1$ and back down", and applying it to hexagon $4$ to predict the row sequence $4,5,6,7,6,5,4$.)
3.NBT.A.2 Fluently add and subtract within 1000 (Adding the seven row counts $4+5+6+7+6+5+4 = 37$ to get the total number of dots in the 4th hexagon.)

⭐ This AMC 8 problem only needs Grade 4 pattern-finding (and a little Grade 3 addition) you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 4)

More like this