O'Halloran numbers: Difference between revisions
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For this task, |
For this task, for our purposes, a cuboid is a 3 dimensional object, with six rectangular faces, where all angles are right angles, opposite faces of the cuboid are equal, and where each dimension is a positive integer unit length. It will subsequently be referred to simply as a cuboid; but be aware that it references the above definition. |
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The surface area of a cuboid |
The surface area of a cuboid is two times the length times the width, plus two times the length times the height, plus two times the width times the height. A cuboid will always have an even integer surface area. The minimum surface area a cuboid may have is 6; one where the '''l''', '''w''', and '''h''' measurements are all '''1'''. |
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2 × ( l × w + w × h + h × l ) |
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Different cuboid configurations (may) yield different surface areas, but the surface area is always an integer and is always even. |
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The minimum surface area a cuboid may have is 6 - namely one for which the '''l''', '''w''', and '''h''' measurements are all 1: |
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A cuboid with l = 2, w = 1 h = 1 has a surface area of 10 |
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2 × ( 2 × 1 + 1 × 1 + 1 × 2 ) = 10 |
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Notice that the total surface area of a cuboid is always an integer and is always even, but the converse is not true. For example, there is no cuboid with a surface area of 8. It is conjectured, however, that for every even integer greater than 924, there is a corresponding cuboid with that area. |
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There is no configuration which will yield a surface area of 8. |
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There are 16 known even integer values below 1000 which can not be a surface area for any integer cuboid. It is conjectured, though not rigorously proved, that no others exist. |
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