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  2. Three utilities problem - Wikipedia

    en.wikipedia.org/wiki/Three_utilities_problem

    A torus allows up to 4 utilities and 4 houses K 3 , 3 {\displaystyle K_{3,3}} is a toroidal graph , which means that it can be embedded without crossings on a torus , a surface of genus one. [ 19 ] These embeddings solve versions of the puzzle in which the houses and companies are drawn on a coffee mug or other such surface instead of a flat ...

  3. Mathematics of paper folding - Wikipedia

    en.wikipedia.org/wiki/Mathematics_of_paper_folding

    The fold-and-cut problem asks what shapes can be obtained by folding a piece of paper flat, and making a single straight complete cut. The solution, known as the fold-and-cut theorem, states that any shape with straight sides can be obtained. A practical problem is how to fold a map so that it may be manipulated with minimal effort or movements.

  4. Langley's Adventitious Angles - Wikipedia

    en.wikipedia.org/wiki/Langley's_Adventitious_Angles

    In 2015, an anonymous Japanese woman using the pen name "aerile re" published the first known method (the method of 3 circumcenters) to construct a proof in elementary geometry for a special class of adventitious quadrangles problem. [7] [8] [9] This work solves the first of the three unsolved problems listed by Rigby in his 1978 paper. [5]

  5. How to Solve It - Wikipedia

    en.wikipedia.org/wiki/How_to_Solve_It

    First, you have to understand the problem. [2] After understanding, make a plan. [3] Carry out the plan. [4] Look back on your work. [5] How could it be better? If this technique fails, Pólya advises: [6] "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"

  6. Hilbert's problems - Wikipedia

    en.wikipedia.org/wiki/Hilbert's_problems

    Of the cleanly formulated Hilbert problems, numbers 3, 7, 10, 14, 17, 18, 19, and 20 have resolutions that are accepted by consensus of the mathematical community. Problems 1, 2, 5, 6, [g] 9, 11, 12, 15, 21, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether they resolve the problems.

  7. Optimal solutions for the Rubik's Cube - Wikipedia

    en.wikipedia.org/wiki/Optimal_solutions_for_the...

    There are many algorithms to solve scrambled Rubik's Cubes. An algorithm that solves a cube in the minimum number of moves is known as God's algorithm. A randomly scrambled Rubik's Cube will most likely be optimally solvable in 18 moves (~ 67.0%), 17 moves (~ 26.7%), 19 moves (~ 3.4%) or 16 moves (~ 2.6%) in HTM. [4]

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