AMC 8 · 2002 · #11

Grade 4 geometry-2dcounting

Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Figure-N Growth Sequence

perfect-squarespattern-recognitionsequences-arithmetic pattern-recognitionidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticperfect-squares

📏 Short solution 💡 2 insights 📊 Diagram

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Problem

A sequence of squares is built from unit tiles. Each square's side is one tile longer than the previous square's side. How many more tiles does the 7th square use than the 6th square?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

The picture already gives the first three cases for free, which is the cue for Tool #5 (Look for a Pattern): count the tiles in those small squares and see what rule governs them. Tool #9 (Solve an Easier Related Problem) is what makes Tool #5 honest — we trust the pattern only after verifying it on the n=1, 2, 3 cases we can see and count. Once the rule "n-th square has n² tiles" is locked in, the 6th and 7th squares come from one subtraction. No algebra needed.

1STEP 1

Count the tiles in the drawn squares — they hold 1, 4, 9 tiles.

1st: 1 tile, 2nd: 4 tiles, 3rd: 9 tiles

2STEP 2

The counts 1, 4, 9 are 1², 2², 3², so the n-th square uses n² tiles.

tiles in n-th square = n²

3STEP 3

Apply n²: the 6th square has 36 tiles, the 7th has 49.

6th: 6² = 36, 7th: 7² = 49

4STEP 4

Subtract: 49 - 36 = 13 more tiles, so (C).

49 - 36 = 13 → (C)

Answer

Sanity check by building the 7th square from the 6th. Start with the 6 × 6 square (36 tiles). To grow it into a 7 × 7, add one new row of 7 tiles along the top and one new column of 7 tiles along the right — but the top-right corner tile is shared. Extra tiles = 7 + 7 - 1 = 13, which matches 49 - 36 = 13. The differences between consecutive square counts are 4-1=3, 9-4=5, 16-9=7, …, the odd numbers, so the gap between the 6th and 7th squares is the 6th odd number = 2 · 6 + 1 = 13. Same answer (C).

💡Key takeaway

Count the small cases (1, 4, 9), spot the square-number rule, then subtract: 7² - 6² = 49 - 36 = 13. Answer (C).

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