AMC 10 · 2005 · #11
Grade 6 geometry-3dPick an answer.
AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Everything can be written in terms of the one unknown n, so Tool #4 (Introduce a Variable) drives the solution. Two counts are needed: the total number of little faces and the number of red little faces. The total is easy — n³ cubes with 6 faces each. The red count needs Tool #17 (Visualize Spatial Relationships): the red little faces are exactly the pieces of the big cube's painted skin, so they cover the original surface of 6 faces of area n². Comparing red to total gives a clean fraction tfrac1n; setting it equal to 1/4 pins down n. Tool #3 (Eliminate Possibilities) then matches the value to the answer list.
Count all the little faces
Cutting gives n³ unit cubes, each with 6 faces of its own, so the whole pile holds 6n³ little faces, painted or not.
Count by cubes first, then multiply by the 6 faces each cube carries.
6.EE.A.1Introduce A VariableCount only the red faces
A cut never makes paint. Red is only the big cube's skin: 6 faces, each an n×n square of n² little squares — so 6n² red faces.
Red never appears on a fresh cut, so the red faces are just the original painted surface.
6.G.A.4Visualize Spatial RelationshipsCompare red to total
Red over total is 6n² divided by 6n³, and since 6n³ is n times 6n², everything cancels down to 1/n.
Every unit cube surrenders 6 faces but only the surface donates red, so the red share thins out as tfrac1n.
Every unit cube gives up six faces but only the outer shell carries paint, so the painted share thins out.
▸ Why?
Paint lives on a surface, which grows with the square of the side, while the cubes grow with its cube.
▸ Why?
A fraction whose top and bottom share a factor can be reduced, leaving one clean factor behind.
Solve for n and pick the choice
The red share is one-fourth, so 1/n=1/4; equal unit fractions force equal denominators, giving n=4, which is choice (B).
If two unit fractions are equal, their denominators are equal.
6.RP.A.3Eliminate PossibilitiesCutting adds lots of bare inside faces but no new paint, so the red share is just 1/n — set that equal to 1/4 to get n=4.
- Count all the little faces
- Count only the red faces
- Compare red to total
- Solve for n and pick the choice