AMC 10 · 2005 · #10
Grade 6 geometry-3dPick an answer.
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
Every unit cube has six faces, so the total is six times the volume.
Count by cubes first, then multiply by the 6 faces each cube carries.
6.EE.A.1Introduce A VariableCount only the red faces
The paint is just the original skin, divided into little squares.
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
Comparing them cancels everything but a single reciprocal.
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 red, so the red share thins out as one over n.
▸ Why?
Red 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, and here everything cancels but one factor of n.
Solve for n and pick the choice
Setting that equal to one fourth gives 4, 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 tfrac1n — 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