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In tiling or tessellation problems, there are to be no gaps, nor overlaps. Many of the puzzles of this type involve packing rectangles or polyominoes into a larger rectangle or other square-like shape.
There are significant theorems on tiling rectangles (and cuboids) in rectangles (cuboids) with no gaps or overlaps:
- An a × b rectangle can be packed with 1 × n strips iff n divides a or n divides b.
- de Bruijn's theorem: A box can be packed with a harmonic brick a × a b × a b c if the box has dimensions a p × a b q × a b c r for some natural numbers p, q, r (i.e., the box is a multiple of the brick.)
The study of polyomino tilings largely concerns two classes of problems: to tile a rectangle with congruent tiles, and to pack one of each n-omino into a rectangle.
A classic puzzle of the second kind is to arrange all twelve pentominoes into rectangles sized 3×20, 4×15, 5×12 or 6×10.
Read more about this topic: Packing Problem
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