A128909 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.

1, 8, 20, 15, 50, 27, 71, 22, 39, 57, 125, 34

Offset: 1

Author

Mauro Fiorentini, Apr 23 2007

Keywords

• hard
• more
• nonn

Comments

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 several unit cubes gives another trivial bound: a(n) <= (9n^2 - 12n + 31) / 4.

Examples

			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.
		

References

• Ainley, Stephen, Mathematical Puzzles, Prentice Hall, New York, 1983. p. 81.

Crossrefs

Cf. A005670.