Slothouber–Graatsma puzzle
Encyclopedia
The Slothouber–Graatsma puzzle is a packing problem that calls for packing six 1 × 2 × 2 blocks and three 1 × 1 × 1 blocks into a 3 × 3 × 3 box. The solution to this puzzle is unique (up to mirror reflections and rotations).
The puzzle is essentially the same if the three 1 × 1 × 1 blocks are left out, so that the task is to pack six 1 × 2 × 2 blocks into a cubic box with volume 27. The Slothouber–Graatsma puzzle is regarded as the smallest nontrivial
3D
packing problem
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rectangular blocks. More complex puzzles involving the packing of convex rectangular blocks have been designed. The best known example is the Conway puzzle
which asks for the packing of eighteen convex rectangular blocks into a 5 x 5 x 5 box. A harder convex rectangular block packing problem is to pack forty-one 1 x 2 x 4 blocks into a 7 x 7 x 7 box (thereby leaving 15 holes).
The puzzle is essentially the same if the three 1 × 1 × 1 blocks are left out, so that the task is to pack six 1 × 2 × 2 blocks into a cubic box with volume 27. The Slothouber–Graatsma puzzle is regarded as the smallest nontrivial
Nontrivial
Nontrivial is the opposite of trivial. In contexts where trivial has a formal meaning, nontrivial is its antonym.It is a term common among communities of engineers and mathematicians, to indicate a statement or theorem that is not obvious or easy to prove.-Examples:*In mathematics, it is often...
3D
Three-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...
packing problem
Packing problem
Packing problems are a class of optimization problems in mathematics which involve attempting to pack objects together , as densely as possible. Many of these problems can be related to real life packaging, storage and transportation issues...
.
Solution
The solution of the Slothouber–Graatsma puzzle is straightforward when one realizes that the three 1 × 1 × 1 blocks (or the three holes) need to be placed along a body diagonal of the box, as each of the 3 x 3 layers in the various directions needs to contain such a unit block. This follows because the larger blocks can only fill an even number of the 9 cells in each 3 x 3 layer.Variations
The Slothouber–Graatsma puzzle is an example of a cube-packing puzzle using convexConvex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
rectangular blocks. More complex puzzles involving the packing of convex rectangular blocks have been designed. The best known example is the Conway puzzle
Conway puzzle
Conway's puzzle is a packing problem using rectangular blocks, named after its inventor, mathematician John Conway. It calls for packing thirteen 1 × 2 × 4 blocks, one 2 × 2 × 2 block, one 1 × 2 × 2 block, and three 1 × 1 × 3 blocks into a 5 × 5 × 5 box....
which asks for the packing of eighteen convex rectangular blocks into a 5 x 5 x 5 box. A harder convex rectangular block packing problem is to pack forty-one 1 x 2 x 4 blocks into a 7 x 7 x 7 box (thereby leaving 15 holes).