Journal for Geometry and Graphics, Vol. 5, No. 1, pp. 1-11 (2001)

# Packing Congruent Bricks into a Cube

## Akos G. Horvath, Istvan Prok

Department of Geometry,
Budapest University of Technology and Economics
Egry Jozsef ut 1, H.22, H-1111 Budapest XI, Hungary
email: ghorvath@math.bme.hu, prok@math.bme.hu

Abstract: L. Lovasz raised the problem whether 27 congruent bricks of edge lengths $a$, $b$, $c$ ($0<a<b<c$, $a+b+c=s$) can be packed into a cube of edge length $s$ without overlaps so that the arrangement is universal, in other words, it should be independent from the choice of $a$, $b$ and $c$. If that were possible, we could obtain a geometric proof of the inequality $\frac 13(a+b+c)\geq\root3\of{abc}$ between the arithmetic and geometric means of three positive numbers. (This would be an analogous method to the well-known proof of the inequality $\frac 12(a+b)\geq\sqrt{ab}$, ($a,b>0$), concerning the packing or four rectangles of edge lengths $a, b$ into a square of edge length $a+b$.) Hence, fundamentally, this is a special packing problem: some bricks having fixed volume must be put into a container of given volume. From the combinatorial point of view, similar container problems were investigated by D. Jennings. The first author has found a possible universal arrangement, and someone else has found an additional one which has proved to be different under the symmetries of the cube. In the paper we introduce an algorithm for finding all the different universal arrangements. As a result we obtain 21 possibilities (listed in Section 4) by the corresponding computer program. Our method seems to be suitable for solving the analogous problem in higher dimensions.

Keywords: packing problem, classification problem

Classification (MSC2000): 52C17

Full text of the article will be available in end of 2002.

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