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The variant where variables are required to be 0 or 1, called zero-one linear programming, and several other variants are also NP-complete [2] [3]: MP1 Some problems related to Job-shop scheduling Knapsack problem , quadratic knapsack problem , and several variants [ 2 ] [ 3 ] : MP9
The Hardest Logic Puzzle Ever is a logic puzzle so called by American philosopher and logician George Boolos and published in The Harvard Review of Philosophy in 1996. [1] [2] Boolos' article includes multiple ways of solving the problem.
Anagrams (also published under names including Anagram, Snatch and Word Making and Taking) is a tile-based word game that involves rearranging letter tiles to form words. The game pieces are a set of tiles with letters on one side.
[1] Can one find a simple closed quasigeodesic on a convex polyhedron in polynomial time? [2] Can a simultaneous embedding with fixed edges for two given graphs be found in polynomial time? [3] Can the square-root sum problem be solved in polynomial time in the Turing machine model?
A Sudoku starts with some cells containing numbers (clues), and the goal is to solve the remaining cells. Proper Sudokus have one solution. [1] Players and investigators use a wide range of computer algorithms to solve Sudokus, study their properties, and make new puzzles, including Sudokus with interesting symmetries and other properties.
Add the clues together, plus 1 for each "space" in between. For example, if the clue is 6 2 3, this step produces the sum 6 + 1 + 2 + 1 + 3 = 13. Subtract this number from the total available in the row (usually the width or height of the puzzle). For example, if the clue in step 1 is in a row 15 cells wide, the difference is 15 - 13 = 2.
If the sum is zero, that subset is a proof or witness for the answer is "yes". An algorithm that verifies whether a given subset has sum zero is a verifier. Clearly, summing the integers of a subset can be done in polynomial time, and the subset sum problem is therefore in NP. The above example can be generalized for any decision problem.
For instance, for a word made of only two different letters, say n letters A and m letters B, the answer is, of course, 1 or 0 according to whether n = m or not, for the only way to form an anagram without fixed letters is to exchange all the A with B, which is possible if and only if n = m.