AMC 10 · 2010 · #17

Grade 7 geometry-3d
Archetype Solid Volume by Decomposition └ Sub-archetype Fraction or Difference of a Bounding Solid
volume-rectangular-prismprinciple-of-inclusion-exclusionspatial-visualization complementary-counting ↑ Prerequisites: volume-rectangular-prism
📏 Medium solution 💡 3 insights
Problem
A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?

Pick an answer.

(A)
7
(B)
8
(C)
10
(D)
12
(E)
15

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Change Focus / Count the Complement

The leftover solid is an awkward frame shape, but the material that was removed is made of simple boxes. So instead of measuring what stays, measure what is taken away and subtract it from the full cube. The catch is that the three tunnels overlap, so counting the removed boxes is its own subproblem that needs care about the shared center.

1STEP 1

Volume of the full cube

Before any drilling, the whole cube of side 3 has volume 27.

3 × 3 × 3 = 27
2STEP 2

Volume of one straight tunnel

One hole is a 2-by-2 square running through the full depth 3, a box of volume 12.

2 × 2 × 3 = 12
3STEP 3

Three tunnels share one center

One tunnel per pair of opposite faces gives 3 tunnels, and all three share the central 2-by-2-by-2 cube of volume 8.

2 × 2 × 2 = 8
4STEP 4

Add up the removed material correctly

Inclusion-exclusion: three tunnels of 12, minus the three overlaps of 8, plus the center 8 back, removes 20 in all.

3 × 12 - 3 × 8 + 8 = 36 - 24 + 8 = 20
5STEP 5

Subtract to get what remains

What is left is the cube minus everything drilled away: 27 minus 20 is 7 cubic inches, choice (A).

27 - 20 = 7
Answer
7
Build the leftover frame directly and check it matches. What survives is 8 corner cubes, each 0.5 by 0.5 by 0.5 (volume 0.125, total 1), plus 12 edge bars, each 2 by 0.5 by 0.5 (volume 0.5, total 6). Adding gives 1 + 6 = 7, the same as 27 minus 20. Also, 7 is the smallest choice, which fits a cube that has been hollowed out almost completely, leaving only a thin frame.
💡Key takeaway

When holes overlap, don't just add up what you drill out; count the shared part once so you don't erase too much.

  • Volume of the full cube
  • Volume of one straight tunnel
  • Three tunnels share one center
  • Add up the removed material correctly
  • Subtract to get what remains

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