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The Slothouber–Graatsma puzzle is a
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 from parity considerations, because the larger blocks can only fill an even number of the 9 cells in each 3 x 3 layer.Elwyn R. Berlekamp, John H. Conway and Richard K. Guy: Winning ways for your mathematical plays, 2nd ed, vol. 4, 2004.
The Slothouber-Graatsma puzzle in Stewart Coffin's "The Puzzling World of Polyhedral Dissections" William Graatsma and Jan Slothouber: Dutch mathematical art
{{DEFAULTSORT:Slothouber-Graatsma puzzle Packing problems Recreational mathematics Tiling puzzles Mechanical puzzle cubes
The Slothouber–Graatsma puzzle is a packing problem
Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few cont ...
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 Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
mirror reflections and rotations). It was named after its inventors Jan Slothouber and William Graatsma.
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.
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 from parity considerations, because the larger blocks can only fill an even number of the 9 cells in each 3 x 3 layer.Elwyn R. Berlekamp, John H. Conway and Richard K. Guy: Winning ways for your mathematical plays, 2nd ed, vol. 4, 2004.
Variations
The Slothouber–Graatsma puzzle is an example of a cube-packing puzzle usingconvex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
polycubes. More general puzzles involving the packing of convex rectangular blocks exist. The best known example is the Conway puzzle
Conway's puzzle, or blocks-in-a-box, 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 ...
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 solution is analogous to the 5x5x5 case, and has three 1x1x5 cuboidal holes in mutually perpendicular directions covering all 7 slices.
See also
*Soma cube
The Soma cube is a solid dissection puzzle invented by Danish polymath Piet Hein in 1933 during a lecture on quantum mechanics conducted by Werner Heisenberg.
Seven pieces made out of unit cubes must be assembled into a 3×3×3 cube. The pie ...
* Bedlam cube
* Diabolical cube
References
External links
The Slothouber-Graatsma puzzle in Stewart Coffin's "The Puzzling World of Polyhedral Dissections"
{{DEFAULTSORT:Slothouber-Graatsma puzzle Packing problems Recreational mathematics Tiling puzzles Mechanical puzzle cubes