benchmarks.wiki / Public workspace
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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 beam is a 1×1×2020 rectangular prism. Several beams are placed…
Problem
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problem
An empty cube is given, and a grid of square unit cells is drawn on each of its six faces. A beam is a rectangular prism. Several beams are placed inside the cube subject to the following conditions:
The two faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are possible positions for a beam.) No two beams have intersecting interiors. The interiors of each of the four faces of each beam touch either a face of the cube or the interior of the face of another beam.
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 beam is a 1×1×2020 rectangular prism. Several beams are placed inside the cube subject to the following conditions: The two faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are possible positions for a beam.) No two beams have intersecting interiors. The interiors of each of the four faces of each beam touch either a face of the cube or the interior of the face of another beam. 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 beam is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions: The two faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are possible positions for a beam.) No two beams have intersecting interiors. The interiors of each of the four faces of each beam touch either a face of the cube or the interior of the face of another beam. What is the smallest positive number of beams that can be placed to satisfy these conditions?
Discussion
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Source and history
- task_id
c179ab40-3cc4-5835-b09b-39cf32a47d06- task_key
test--c179ab40-3cc4-5835-b09b-39cf32a47d06- task_revision_id
- 1
- config / split
- / test
initial import