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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.
The most common problem being solved is the 0-1 knapsack problem, which restricts the number of copies of each kind of item to zero or one. Given a set of items numbered from 1 up to , each with a weight and a value , along with a maximum weight capacity ,
The goal is to pack as many small rectangles as possible into the big rectangle without overlap between any rectangles (small or large). Common constraints of the problem include limiting small rectangle rotation to 90° multiples and requiring that each small rectangle is orthogonal to the large rectangle.
It requires Θ(n log n) time, where n is the number of items to be packed. The algorithm can be made much more effective by first sorting the list of items into decreasing order (sometimes known as the first-fit decreasing algorithm), although this still does not guarantee an optimal solution and for longer lists may increase the running time ...
Indiana's alcohol laws have changed a few times over the years, most recently this year, when the happy hour ban was lifted. Here's what to know.
This population of athletes is a prime example of a strong core that, at times, don't have the traditional sculpted six-pack abs. So yes, it is possible to have a strong and healthy core without ...
Square packing in a circle is a related problem of packing n unit squares into a circle with radius as small as possible. For this problem, good solutions are known for n up to 35. Here are the minimum known solutions for up to n=12: [11] (Only the cases n=1 and n=2 are known to be optimal)
Men's Health experts Dave Rienzi and Dr. Pat Davidson break down one of the holy grails of fitness goals: body recomposition.