AMC 10 · 2012 · #22
Grade 10 geometry-3dPick an answer.
The condition is two-sided: no plane may mark anything outside the given segments, and the planes together must mark all of them. Tool #7 (Identify Subproblems) splits this into two independent questions — how many planes are legal at all (that maximum is just the size of the legal list), and how few legal planes still cover every segment. Tool #4 (Introduce a Variable) puts the cube at [-1,1]³ so every segment becomes an equation. The load-bearing step is a lemma from Tool #17 (Visualize Spatial Relationships): where a legal plane crosses a face, the crossing must be one whole marked segment, so the cross-section is a polygon built out of marked segments and its corners are edge midpoints. That turns an infinite search over planes into a finite one, which Tool #2 (Make a Systematic List) closes by sorting the planes by the direction they face. Tool #14 (Extreme Principle) supplies the lower bound for the minimum: count how much each type of plane can possibly cover and show the budget forbids doing it with six.
Put the cube on coordinates
Coordinates make the 36 marks explicit.
Every marked segment ends at an edge midpoint, so coordinates with one zero and two ± 1's describe the whole picture.
10.G-CO.A.1Introduce A VariableA legal cut only follows whole traces
A cut's outline is built of whole marks.
A straight chord has to lie on one of the drawn lines, and on a face each drawn line is exactly one of the marked segments.
7.G.A.3Visualize Spatial RelationshipsOnly 13 directions are possible
Only 13 directions are possible at all.
The sides of the cut are made of traces, so the plane is pinned down by the few directions the traces actually run in.
10.G-GPE.B.4Visualize Spatial RelationshipsCount the planes in each direction
Counting per direction gives 27 planes in all.
Fix the direction a plane faces and only a couple of positions leave a legal mark, so the whole list can be written out.
7.G.A.3Make A Systematic ListSeven planes already cover everything
Just 7 of them already cover everything.
Planes through the center are the greedy choice — they cut all the way across, so each one sweeps up the most segments.
10.G-CO.A.3Identify SubproblemsSix planes cannot be enough
A counting bound shows six cannot suffice.
Give each plane a covering budget: the best plane covers six segments, so thirty-six segments need at least six planes — and there are not six of the best kind.
Give each plane a covering budget: the best plane covers six segments, so thirty-six segments need at least six planes.
▸ Why?
If fewer planes than that were used, some segment would be left with no plane covering it.
▸ Why?
No plane can beat that budget, so the total covered can never exceed the planes times six.
Take the difference
The spread is 20, choice (C).
One count says how many cuts are allowed, the other says how few are needed, and the question asks only for the gap.
7.EE.B.4Identify SubproblemsA cut can only leave a mark along lines that are already drawn, so first list every plane that is allowed — there are 27 — then find the fewest of them that still draw every mark, which is the 7 passing through the center.
- Put the cube on coordinates
- A legal cut only follows whole traces
- Only 13 directions are possible
- Count the planes in each direction
- Seven planes already cover everything
- Six planes cannot be enough
- Take the difference
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