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Search: id:A128909
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| A128909 |
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3D version of A005670. The problem is to dissect an n X n X n cube into smaller integer cubes, the gcd of whose sides is 1, using the smallest number of cubes. The gcd condition exclude dissecting a 6 X 6 X 6 cube into eight 3 X 3 X 3 cubes. |
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+0
1
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| 1, 8, 20, 15, 50, 27, 71, 22, 39, 57, 125, 34
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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As far as I know, no term, (except trivial cases) has been proved optimal. Repeated dissection, as in the above example, shows that if the side is a composite number mn, a(mn) <= a(m) + a(n) - 1. It is an open problem to find a number mn for which a(mn) < a(m) + a(n) - 1. Dissecting a cube with side n into a cube with side n - 1 and several unit cubes gives a trivial bound: a(n) <= 3n^2 - 3n + 2. Dissecting a cube with side n = 2k + 1 into a cube with side k + 1, 7 with side k and and several unit cubes gives another trivial bound: a(n) <= (9n^2 - 12n + 31) / 4.
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REFERENCES
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Ainley, Stephen, Mathematical Puzzles, Prentice Hall, New York, 1983. p. 81.
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EXAMPLE
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a(4)=15 because a 4 X 4 X 4 cube can be dissected into 8 2 X 2 X 2, one of which can be dissected into 8 1 X 1 X 1.
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CROSSREFS
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Cf. A005670.
Sequence in context: A034433 A120081 A081963 this_sequence A115147 A022700 A100212
Adjacent sequences: A128906 A128907 A128908 this_sequence A128910 A128911 A128912
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KEYWORD
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hard,more,nonn
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AUTHOR
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Mauro Fiorentini (mauro.fiorentini(AT)fastwebnet.it), Apr 23 2007
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