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The missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures; or rather to teach them not to reason using figures, but to use only textual descriptions and the axioms of geometry. It depicts two arrangements made of similar shapes in slightly different configurations.
Chessboard paradox. The chessboard paradox [1] [2] or paradox of Loyd and Schlömilch [3] is a falsidical paradox based on an optical illusion. A chessboard or a square with a side length of 8 units is cut into four pieces. Those four pieces are used to form a rectangle with side lengths of 13 and 5 units.
(C) Berry's Paradox with letters instead of syllables is often related to the set of all natural numbers which can be defined by less than 100 (or any other large number) letters. As the natural numbers are a well-ordered set there must be the least number which cannot be defined by less than 100 letters .
The total number of grains can be shown to be 2 64 −1 or 18,446,744,073,709,551,615 (eighteen quintillion, four hundred forty-six quadrillion, seven hundred forty-four trillion, seventy-three billion, seven hundred nine million, five hundred fifty-one thousand, six hundred and fifteen, over 1.4 trillion metric tons), which is over 2,000 times ...
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Hooper's paradox is a falsidical paradox based on an optical illusion. A geometric shape with an area of 32 units is dissected into four parts, which afterwards get assembled into a rectangle with an area of only 30 units.
Proebsting's paradox apparently shows that the Kelly criterion can lead to ruin. Sleeping Beauty problem: A probability problem that can be correctly answered as one half or one third depending on how the question is approached. Three Prisoners problem, also known as the Three Prisoners paradox: [3] A variation of the Monty Hall problem.
The solutions in both cases are non-trivial but yield to straightforward application of trigonometry, analytical geometry or integral calculus. Both problems are intrinsically transcendental – they do not have closed-form analytical solutions in the Euclidean plane. The numerical answers must be obtained by an iterative approximation procedure.