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The block-stacking problem is the following puzzle: Place identical rigid rectangular blocks in a stable stack on a table edge in such a way as to maximize the overhang. Paterson et al. (2007) provide a long list of references on this problem going back to mechanics texts from the middle of the 19th century.
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 solution is analogous to the 5x5x5 case, and has three 1x1x5 cuboidal ...
Pieces used in 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 × 1 × 3 blocks into a 5 × 5 × 5 box.
The blocks world is a planning domain in artificial intelligence. The algorithm is similar to a set of wooden blocks of various shapes and colors sitting on a table. The goal is to build one or more vertical stacks of blocks. Only one block may be moved at a time: it may either be placed on the table or placed atop another block.
The agent must stack the blocks such that A is atop B, which in turn is atop C. However, it may only move one block at a time. The problem starts with B on the table, C atop A, and A on the table: However, noninterleaved planners typically separate the goal (stack A atop B atop C) into subgoals, such as: get A atop B; get B atop C
The general problem of solving Sudoku puzzles on n 2 ×n 2 grids of n×n blocks is known to be NP-complete. [8] A puzzle can be expressed as a graph coloring problem. [9] The aim is to construct a 9-coloring of a particular graph, given a partial 9-coloring. The Sudoku graph has 81 vertices, one vertex for each cell.
Famous examples include the number of ways to place n non-attacking rooks on: an entire n × n chessboard, which is an elementary combinatorial problem; the same board with its diagonal squares forbidden; this is the derangement or "hat-check" problem (this is a particular case of the problème des rencontres);
An initial configuration. A solution. Shikaku is played on a rectangular grid. Some of the squares in the grid are numbered. The objective is to divide the grid into rectangular and square pieces such that each piece contains exactly one number, and that number represents the area of the rectangle.