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Omni-MATH / An empty 2020×2020×2020 cube is given, and a 2020×2020 grid of square unit cells is drawn on each of its six faces. A [i]beam[/i] is a 1×1×2020 rectangular prism. Several beams are…

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problem

An empty 2020×2020×20202020 \times 2020 \times 2020 cube is given, and a 2020×20202020 \times 2020 grid of square unit cells is drawn on each of its six faces. A [i]beam[/i] is a 1×1×20201 \times 1 \times 2020 rectangular prism. Several beams are placed inside the cube subject to the following conditions: [list=] [*]The two 1×11 \times 1 faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are 3⋅202023 \cdot {2020}^2 possible positions for a beam.) [*]No two beams have intersecting interiors. [*]The interiors of each of the four 1×20201 \times 2020 faces of each beam touch either a face of the cube or the interior of the face of another beam. [/list] What is the smallest positive number of beams that can be placed to satisfy these conditions?
Plain-text mathematical notation (without MathML)
An empty 2020×2020×2020 cube is given, and a 2020×2020 grid of square unit cells is drawn  on each of its six faces. A [i]beam[/i] is a 1×1×2020 rectangular prism. Several beams are placed inside the cube subject to the following conditions:
[list=]
[*]The two 1×1 faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are 3⋅2020² possible positions for a beam.)
[*]No two beams have intersecting interiors.
[*]The interiors of each of the four 1×2020 faces of each beam touch either a face of the cube or the interior of the face of another beam.
[/list]
What is the smallest positive number of beams that can be placed to satisfy these conditions?
Original LaTeX notation
An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn  on each of its six faces. A [i]beam[/i] is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:
[list=]
[*]The two $1 \times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \cdot {2020}^2$ possible positions for a beam.)
[*]No two beams have intersecting interiors.
[*]The interiors of each of the four $1 \times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam.
[/list]
What is the smallest positive number of beams that can be placed to satisfy these conditions?

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Source and history

task_id
e603e40e-b4f8-50de-a741-6a0f9aaeae6d
task_key
test--e603e40e-b4f8-50de-a741-6a0f9aaeae6d
task_revision_id
1
config / split
/ test

Official source

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