The Slotober – Graatsma puzzle is the task of packing six 1 × 2 × 2 blocks and three 1 × 1 × 1 blocks into 3 × 3 × 3 cube. The solution to the puzzle is unique (accurate to mirror reflections and turns).
The puzzle will essentially be the same if you remove the blocks 1 × 1 × 1 and the task is to pack six 1 × 2 × 2 blocks into a cube of volume 27. The Slotober – Graatsma puzzle is considered as the smallest (known) nontrivial 3D packing problem .
Content
Solution
Solving the Slotober-Graatsma puzzle is simple, if you realize that three 1 × 1 × 1 blocks (or three voids) should be placed along the big diagonal of the cube, since at each level there should be one such block in all directions. This follows from parity considerations, since large blocks can fill an even number of nine cells of each 3 x 3 level [1] .
Variations
The Slotober – Graatsma puzzle is an example of a cube packing puzzle that uses convex polycubes . Other jigsaw puzzles for packing convex rectangular blocks are known. The most famous of them is Conway's puzzle , in which you need to pack eighteen rectangular blocks into a 5 x 5 x 5 cube. The more difficult task is to pack 41 rectangular blocks 1 x 2 x 4 into a 7 x 7 x 7 cube (there are 15 empty cells left) [1] .
See also
- Cubes of catfish
- Bedlam Cube
- Devil's Cube
Notes
- ↑ 1 2 Elwyn R. Berlekamp, John H. Conway and Richard K. Guy: Winning ways for your mathematical plays, 2nd ed, vol. 4, 2004.
Links
- Slotober Puzzle - Gratsma in Stewart Coffin's book, “The Puzzling World of Polyhedral Dissections”
- Slotober and Graatsma: Picking Up the Cube
- [1] Slotober and Graatsma: Danish Mathematical Art]