A cube is divided into 27 equal smaller cubes. A plane intersects the cube. Find the maximum possible number of smaller cubes the plane can intersect.
I can't seem to do better than 17, assuming the plane must intersect the insides of the smaller cubes. (It not then the solution is 18)
First consider the 2-Dimensional case of a line intersecting a 3x3 array of squares. It is easy to see the maximum is 5 squares.
--X
-XX
XX-
Then to extend to 3-dimensions, give the plane a small tilt to hit 7 in the middle layer.
--X -XX -XX
-XX XXX XX-
XX- XX- X--
If hitting the edge of a cube counts, you can slice parallel to a side and hit 6 cubes on each layer:
XX-
XX-
XX-
Edit:
I should have considered angling the plane section more. If it is perpendicular to the space diagonal of the cube, it will hit the corners. (and have more symmetry.) Now 19 of the cubes are cut.
--X -XX XXX
-XX XXX XX-
XXX XX- X--
The only ones missed are 4 in the two opposite corners.
Edited on November 25, 2020, 9:14 am
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Posted by Jer
on 2020-11-24 12:02:44 |
My guess
