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HW 1 - 5 A 3D-septomino is a three-dimensional 2 2 2 × ×...

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1. Show that n n 2 ! for all 4 n by induction. 2. Show that 1 2 ! + n n for all 5 n by induction. n k n 2 ! 3. Use induction to show that n straight lines in the plane, with no two parallel and no three going through a common point, divide the plane into 2 / ) 2 ( 2 + + n n regions. 4. Show that the regions of the preceding exercise can be colored red and green so that no two regions sharing a common edge have the same color.
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Unformatted text preview: 5. A 3D-septomino is a three-dimensional 2 2 2 × × cube with one of the little 1 1 1 × × corner cubes removed. Prove that a n n n 2 2 2 × × cube with any one of the 1 1 1 × × cubes can be tiled by 3D-septominoes. 6. Prove by induction that 1 1 4 3 3 2 2 1 2 + < + + ⋅ ⋅ ⋅ + + + n n n n ....
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