AMC 10 · 2006 · #5

Grade 4 geometry-2d
Archetype Grid Tiling / Packing with Divisibility └ Sub-archetype Fixed Piece Set — Feasibility by Elimination
area-rectanglesspatial-visualizationoptimization guess-and-check ↑ Prerequisites: area-rectangles
📏 Medium solution 💡 2 insights
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Problem
A 2×3 rectangle and a 3×4 rectangle must sit inside one square without overlapping, and every rectangle side stays parallel to the square's sides. Find the smallest possible area of that square.

Pick an answer.

(A)
16
(B)
25
(C)
36
(D)
49
(E)
64

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Extreme Principle

The question asks for a smallest value, so treat it as a minimum problem. First find a lower bound the square's side must clear, then show the smallest allowed size actually holds both rectangles. A diagram and some spatial reasoning confirm that a real placement exists.

1STEP 1

Big rectangle sets a floor

The 3×4 rectangle's longest side is 4, so the square's side must be at least 4; the choices leave sides 4, 5, 6, 7, 8 to test.

side ≥ 4
2STEP 2

Area rules out side 4

Their areas add to 2×3 + 3×4 = 6 + 12 = 18, but a 4×4 square offers only 16 — so side 4 cannot hold both.

2 · 3 + 3 · 4 = 6 + 12 = 18 ＞ 16 = 4 · 4
3STEP 3

Side 5 actually works

In a 5×5 square lay the 3×4 along the bottom, 4 wide and 3 tall; the leftover 5-by-2 top strip takes the 2×3 with no overlap.

3 × 4 (bottom) + 2 × 3 (top strip) ⊂ 5 × 5
4STEP 4

Smallest square found

Side 4 is impossible and side 5 works, so the smallest square has side 5 and area 5×5 = 25. The answer is (B).

5 × 5 = 25
Answer
25
The answer 25 is one of the offered choices and sits just above the impossible 16. The two rectangles cover 18 square units, so any working square needs area at least 18; 25 is the smallest listed square area that clears 18 and also admits a real placement. A side of 5 comfortably contains the longest rectangle side of 4.
💡Key takeaway

First find the size too small to hold everything, then show the next size up really fits, and that next size is the smallest that works.

  • Big rectangle sets a floor
  • Area rules out side 4
  • Side 5 actually works
  • Smallest square found

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