AMC 8 · 2020 · #4

Grade 4 pattern
Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Figure-N Growth Sequence
pattern-recognitionsequences-arithmeticspatial-visualization pattern-recognitionidentify-subproblems ↑ Prerequisites: multi-digit-arithmetic
📏 Medium solution 💡 3 insights 📊 Diagram
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Problem
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?

Pick an answer.

(A)
35
(B)
37
(C)
39
(D)
43
(E)
49

AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

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, … because the row-by-row structure is visually obvious and a Grade 4 pattern-recognition argument is enough.

1STEP 1

Read the three hexagons by rows: hexagon 1 is 1; hexagon 2 is 2, 3, 2; hexagon 3 is 3, 4, 5, 4, 3 — symmetric, longest in the middle.

1; 2,3,2; 3,4,5,4,3
2STEP 2

The rule: the n-th hexagon has 2n-1 rows counting n, n+1, …, 2n-1, …, n+1, n. Check: n=2 gives 2, 3, 2 and n=3 gives 3, 4, 5, 4, 3.

n=2: 2,3,2 ✓ n=3: 3,4,5,4,3 ✓
3STEP 3

Set n=4: the 4th hexagon has 2(4)-1 = 7 rows, so the dot counts are 4, 5, 6, 7, 6, 5, 4 — listed in order, none skipped.

4, 5, 6, 7, 6, 5, 4
4STEP 4

Add them, pairing the symmetric rows: (4+4) + (5+5) + (6+6) + 7 = 8 + 10 + 12 + 7 = 37.

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

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

37 → (B)
Answer
37
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× 3 = 19 + 18 = 37 — matching choice (B). The next term would be 37 + 24 = 61, which is the formula for centered hexagonal numbers 3n² - 3n + 1 evaluated at n=5: a satisfying sanity check.
💡Key takeaway

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

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