AMC 8 · 2024 · #3

Grade 3 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Nested Frame / Concentric Difference

area-rectanglesperfect-squares area-difference ↑ Prerequisites: area-rectanglesmulti-digit-arithmetic

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Four squares with side lengths 4, 7, 9, and 10 are stacked so their left edges and bottom edges all meet at the same corner. From largest to smallest the colors alternate gray–white–gray–white. Find the total area (in square units) of the gray region that is still visible.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The visible gray is not one blob — it is two separate L-shaped bands: an outer band between the side-10 gray and the side-9 white, plus an inner band between the side-7 gray and the side-4 white. Tool #7 (Identify Subproblems) lets us split the question into "find each gray band, then add them." Each band is just (big square area) − (small square area), a one-line calculation. Tool #1 (Draw a Diagram) confirms which gray is visible, and Tool #3 (Eliminate Possibilities) verifies against the multiple-choice list at the end.

1STEP 1

Shared corner means each gray square is covered by one white, leaving two visible gray bands: side-10 under side-9, side-7 under side-4.

visible gray = (outer gray band) + (inner gray band)

2STEP 2

Outer band = big gray minus the white on it: 10 × 10 − 9 × 9 = 100 − 81 = 19.

10 × 10 - 9 × 9 = 100 - 81 = 19

3STEP 3

Same idea for the inner band: side-7 gray minus side-4 white = 7 × 7 − 4 × 4 = 49 − 16 = 33.

7 × 7 - 4 × 4 = 49 - 16 = 33

4STEP 4

The two gray bands do not overlap (each lives between a different pair of squares), so the total visible gray area is simply their sum.

19 + 33 = 52

5STEP 5

Match 19 + 33 = 52 against the choices — only (E) fits, and it sits well below the side-10 square's area of 100.

19 + 33 = 52 → (E)

Answer

Intuition check: the side-10 gray has area 100 and the side-9 white covers 81 of it, so the outer gray band is only 19 — a thin strip, which makes sense because the two side lengths differ by just 1. The inner gray, between side 7 and side 4, has a much bigger side-length gap (3), so its band is fatter (33). Adding 19 + 33 = 52 sits well below the full 100 of the big square and is exactly the sum of two non-overlapping bands. Answer (E) 52 is reasonable.

💡Key takeaway

This AMC 8 problem only needs Grade 3 square-area (side × side) and subtracting-the-inside-from-the-outside that you already know!

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