Cube Cross Sections

Probability Level 4

A plane cuts through a cube and intersects the center point inside the cube. For how many different planes will the intersection of the plane and the surface of the cube form a regular polygon?

6 7 12 9 3 14 8

2 solutions

Maggie Miller
Aug 27, 2015

See Bufang's solution for completeness, but here's an illustration of the 7 regular polygonal cross sections passing through the middle (three squares and four hexagons):

Bufang Liang
Aug 27, 2015

Rather than thinking about the planes themselves, let's consider the vectors that are perpendicular to the plane in which the polygon lies. The vectors, starting from the center of the cube, can point to the center of any face or any vertex, a total of $6+8 = 14$ different choices. We divide our final total by 2 since each plane has 2 vectors that can correspond to it. Our final answer is $\boxed{7}$ .

The main reason for undercounting is probably not realizing the planes whose perpendicular vector point to a vertex of the cube form a regular hexagon.

The main reason for overcounting is probably not realizing that the planes whose perpendicular vector point to the middle of an edge of the cube do not form a regular polygon, but a rectangle.

I did it by visualisation. I could only conceive of 3 squares and 4 hexagons.

$3$ squares like you're putting ribbons on a present.

$4$ hexagons, one for each long diagonal of the cube. How do I know cutting perpendicular though the diagonal of a cube makes a perfect hexagon without too much effort? I've already posted a problem here which illustrates it.

Isaac Buckley - 5 years, 9 months ago

Here's one way: Keep in mind that all of the points on the cross section (the hexagon) are equal distance to the two opposite vertices of the cube. Label a few of the points you can find until you are able to visualize the edge of the hexagon.

Bufang Liang - 1 year, 3 months ago

Cube Cross Sections

2 solutions

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