{"license":"CC BY-SA 4.0","total":286,"page":1,"pages":6,"problems":[{"slug":"Needles_tip","url":"https://problempedia.org/p/Needles_tip","title":"Needle's tip","statement":"Estimate the size of a needle's tip.","field":"physics","tags":["everyday-physics"],"difficulty":1,"difficulty_from":"solvers","origin":"original","source":null,"lang":"en","author":"igor","solutions":1,"likes":1,"created_at":"2026-10-03T14:55:27Z","updated_at":"2026-10-04T18:40:27Z"},{"slug":"Feynmans_reverse_sprinkler","url":"https://problempedia.org/p/Feynmans_reverse_sprinkler","title":"Feynman's reverse sprinkler","statement":"If a lawn sprinkler is submerged and made to suck water in instead of spray it out, which way does it spin?","field":"physics","tags":["fluid-dynamics"],"difficulty":null,"difficulty_from":null,"origin":"archive","source":"Ernst Mach (1883); popularized by Richard Feynman's inconclusive Princeton cyclotron-lab experiment (c. 1940)","lang":"en","author":"archive","solutions":0,"likes":0,"created_at":"2026-10-01T18:42:00Z","updated_at":"2026-10-01T18:42:00Z"},{"slug":"The_Collatz_conjecture","url":"https://problempedia.org/p/The_Collatz_conjecture","title":"The Collatz conjecture","statement":"Start with any positive integer $n$. 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Thomson/Bowden pressure-melting hypothesis (1930s)","lang":"en","author":"archive","solutions":0,"likes":0,"created_at":"2026-10-01T14:13:22Z","updated_at":"2026-10-01T14:13:22Z"},{"slug":"P_versus_NP","url":"https://problempedia.org/p/P_versus_NP","title":"P versus NP","statement":"P is the class of decision problems that can be solved in polynomial time, and NP is the class of decision problems whose yes answers can be checked in polynomial time given a suitable certificate. Is P equal to NP? 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Is there an algorithm that does this in time polynomial in $n$?","field":"cs","tags":["complexity-theory","graph-algorithms","open-question"],"difficulty":5,"difficulty_from":"author","origin":"archive","source":"folklore","lang":"en","author":"archive","solutions":0,"likes":0,"created_at":"2026-09-30T02:12:53Z","updated_at":"2026-09-30T02:12:53Z"},{"slug":"The_teapot_effect","url":"https://problempedia.org/p/The_teapot_effect","title":"The teapot effect","statement":"Why does liquid poured slowly from a spout often dribble down the outside of the spout instead of falling cleanly into the cup?","field":"physics","tags":["everyday-physics","fluid-dynamics"],"difficulty":null,"difficulty_from":null,"origin":"archive","source":"Markus Reiner, who coined the term (1956); the phenomenon was known to tea-drinkers and potters long before","lang":"en","author":"archive","solutions":0,"likes":0,"created_at":"2026-09-30T00:38:35Z","updated_at":"2026-09-30T00:38:35Z"},{"slug":"Goldbachs_conjecture","url":"https://problempedia.org/p/Goldbachs_conjecture","title":"Goldbach's conjecture","statement":"Prove that every even integer greater than 2 is the sum of two primes. 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The schoolbook method shows $\\omega \\le 3$, and $\\omega \\ge 2$ because the output has $n^2$ entries. Is $\\omega = 2$?","field":"cs","tags":["algebraic-complexity","algorithms","linear-algebra","open-question"],"difficulty":5,"difficulty_from":"author","origin":"archive","source":"Volker Strassen, 1969","lang":"en","author":"archive","solutions":0,"likes":0,"created_at":"2026-09-27T21:56:01Z","updated_at":"2026-09-27T21:56:01Z"},{"slug":"Ramanujans_nested_radical","url":"https://problempedia.org/p/Ramanujans_nested_radical","title":"Ramanujan's nested radical","statement":"Find the value of $$\\sqrt{1 + 2\\sqrt{1 + 3\\sqrt{1 + 4\\sqrt{1 + \\cdots}}}}$$ and explain in what sense the infinite expression has a value.","field":"math","tags":["algebra","sequences"],"difficulty":3,"difficulty_from":"author","origin":"archive","source":"Srinivasa Ramanujan, 1911","lang":"en","author":"archive","solutions":0,"likes":1,"created_at":"2026-09-27T11:48:36Z","updated_at":"2026-09-27T11:48:36Z"},{"slug":"Coffee-ring_effect","url":"https://problempedia.org/p/Coffee-ring_effect","title":"Coffee-ring effect","statement":"Why does a drying drop of coffee (or any particle-laden liquid) leave a dense ring stain at its edge rather than a uniform spot?","field":"physics","tags":["everyday-physics","fluid-dynamics"],"difficulty":null,"difficulty_from":null,"origin":"archive","source":"common household observation, 1990","lang":"en","author":"archive","solutions":0,"likes":0,"created_at":"2026-09-27T00:53:10Z","updated_at":"2026-09-27T00:53:10Z"},{"slug":"The_sum_of_the_first_n_cubes","url":"https://problempedia.org/p/The_sum_of_the_first_n_cubes","title":"The sum of the first n cubes","statement":"Prove that for every positive integer $n$ $$1^3 + 2^3 + \\cdots + n^3 = (1 + 2 + \\cdots + n)^2.$$","field":"math","tags":["algebra","induction"],"difficulty":1,"difficulty_from":"author","origin":"archive","source":"Nicomachus of Gerasa, around 100 AD","lang":"en","author":"archive","solutions":0,"likes":0,"created_at":"2026-09-25T21:07:30Z","updated_at":"2026-09-25T21:07:30Z"},{"slug":"The_chromatic_number_of_the_plane_(Hadwiger-Nelson_problem)","url":"https://problempedia.org/p/The_chromatic_number_of_the_plane_(Hadwiger-Nelson_problem)","title":"The chromatic number of the plane (Hadwiger–Nelson problem)","statement":"Color every point of the plane so that any two points at distance exactly 1 receive different colors. 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Prove that for each runner there is a moment when every other runner is at distance at least $1/n$ along the track. The speeds $0, 1, \\dots, n-1$ show that $1/n$ cannot be replaced by anything larger.","field":"math","tags":["combinatorics","diophantine-approximation","number-theory","open-question"],"difficulty":5,"difficulty_from":"author","origin":"archive","source":"Jörg M. 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Is every problem in BPP solvable by a deterministic polynomial-time algorithm, that is, is P = BPP?","field":"cs","tags":["complexity-theory","derandomization","open-question","randomized-algorithms"],"difficulty":5,"difficulty_from":"author","origin":"archive","source":"folklore","lang":"en","author":"archive","solutions":0,"likes":0,"created_at":"2026-09-15T11:22:33Z","updated_at":"2026-09-15T11:22:33Z"},{"slug":"A_symmetric_system_of_three_equations","url":"https://problempedia.org/p/A_symmetric_system_of_three_equations","title":"A symmetric system of three equations","statement":"Solve the system $$x + y + z = 6,\\qquad xy + yz + zx = 11,\\qquad xyz = 6.$$","field":"math","tags":["algebra","polynomials"],"difficulty":1,"difficulty_from":"author","origin":"archive","source":"School algebra","lang":"en","author":"archive","solutions":0,"likes":0,"created_at":"2026-09-14T10:09:52Z","updated_at":"2026-09-14T10:09:52Z"},{"slug":"Splash_and_crown_formation_of_a_falling_drop","url":"https://problempedia.org/p/Splash_and_crown_formation_of_a_falling_drop","title":"Splash and crown formation of a falling drop","statement":"Why does a liquid drop hitting a dry surface sometimes splash into a crown of droplets and sometimes spread out smoothly instead?","field":"physics","tags":["everyday-physics","fluid-dynamics","partly-solved"],"difficulty":null,"difficulty_from":null,"origin":"archive","source":"A. 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A simple algorithm takes $\\mathcal{O}(n^2)$ time. Is there an algorithm that runs in time $\\mathcal{O}(n^{2-\\varepsilon})$ for some fixed $\\varepsilon > 0$?","field":"cs","tags":["algorithms","computational-geometry","fine-grained-complexity","open-question"],"difficulty":5,"difficulty_from":"author","origin":"archive","source":"A. Gajentaan and M. H. 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