AMC 10 · 2022 · #3

Grade 6 geometry-2d
Archetype Grid Tiling / Packing with Divisibility └ Sub-archetype Fixed Piece Set — Feasibility by Elimination
area-rectanglesperfect-squaresperimeterspatial-visualization complementary-countingidentify-subproblemsconvert-to-algebra ↑ Prerequisites: area-rectangles
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
Five rectangles tile a square with no gaps and no overlaps. Four sit in the four corners and the fifth sits shaded in the middle. The five sizes are given and named, but the picture is not drawn to scale. Decide which named rectangle is the shaded middle one.

Pick an answer.

(A)
A
(B)
B
(C)
C
(D)
D
(E)
E
How to solve
Strategy Change Focus / Count the Complement

Chasing where each rectangle goes means guessing four placements at once, and the picture is not to scale so it gives no hint. Tool #16 (Change Focus / Count the Complement) flips the question: instead of locating the middle rectangle, measure everything the four corner rectangles use up, then take the middle one as what is left over. Tool #7 (Identify Subproblems) splits the work into two easy counts — total area, which gives the side length, and the total of every length plus width. Tool #4 (Introduce a Variable) names the cut positions to prove the corner total is fixed no matter where the cuts fall. Tool #3 (Eliminate Possibilities) then tests the five candidates against that leftover number, and only one survives.

1STEP 1

Add the five areas

Adding the areas gives the size.

6+8+30+14+6=64
2STEP 2

Turn area into side length

The square root gives the side.

n × n=64 → n=8, perimeter=4×8=32
3STEP 3

Read the four sides

The corner rectangles fill the perimeter.

Σ₄ corner rectangles(length+width)=perimeter=32
4STEP 4

Check the total with letters

Writing it in letters confirms the total.

(p+q)+(r+8-q)+(8-r+8-t)+(8-p+t)=32
5STEP 5

Match the leftover

The leftover matches the two by four rectangle.

38-32=6, 2+4=6 → (B)
Answer
B
Only one of the five candidates has length plus width equal to 6, so nothing else could have worked. An actual arrangement also exists, which confirms the reasoning is not empty. With side 8, take the cuts at p=5, q=6, r=7, t=2: the bottom-left rectangle is 5×6 (that is C), the top-left is 7×2 (that is D), the top-right is 1×6 (that is A), the bottom-right is 3×2 (that is E), and the middle is (7-5)×(6-2)=2×4 (that is B). The areas check out too: 30+14+6+6+8=64. Note the middle rectangle comes out taller than it is wide, unlike the picture, which is exactly why the problem warns that the figure is not to scale.
💡Key takeaway

The four border rectangles' lengths and widths are exactly the pieces of the square's perimeter, so they always add to 32 — subtract that from the total 38 and the middle rectangle has to be the one with length plus width equal to 6.

  • Add the five areas
  • Turn area into side length
  • Read the four sides of the square
  • Check the total with letters
  • Subtract, then match the leftover

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