A vanishing puzzle is a mechanical optical illusion comprising multiple pieces which can be rearranged to show different versions of a picture depicting several objects, the number of which depending on the arrangement of the pieces. [1] [2]
A vanishing puzzle is a mechanical optical illusion comprising multiple pieces which can be rearranged to show different versions of a picture depicting several objects, the number of which depending on the arrangement of the pieces. [1] [2]
Wemple & Company marketed an advertising card named The Magic Egg Puzzle, (How Many Eggs?) in New York in 1880. [3] Cutting the rectangular card into four oblongs allowed the pieces to be rearranged to show either 8, 9 or 10 eggs. Many other similar puzzles have been published since. [4]
Chess player and recreational mathematician Sam Loyd patented rotary vanishing puzzles in 1896 and published versions named Get Off the Earth, Teddy and the Lion and The Disappearing Bicyclist (pictured). Each had a circular card connected to a cardboard backdrop with a pin, letting it freely rotate. [5] [6] [7] In The Disappearing Bicyclist, when the disc is rotated such that the arrow points to A, 13 boys can be counted, but when it points to B, there are only 12 boys. [8]
Prizes from $5 to $100 were offered for the best explanation of one illusion. Though the names of the winners were published, their explanations were not. [9]
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. Each apparently forms a 13×5 right-angled triangle, but one has a 1×1 hole in it.
Sam Loyd's chessboard paradox demonstrates two rearrangements of an 8×8 square. In the "larger" rearrangement (the 5×13 rectangle in the image to the right), the gaps between the figures have a combined unit square more area than their square gaps counterparts, creating an illusion that the figures there take up more space than those in the original square figure. [10]
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research- and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited to being an endeavor for amateurs, many topics in this field require no knowledge of advanced mathematics. Recreational mathematics involves mathematical puzzles and games, often appealing to children and untrained adults and inspiring their further study of the subject.
Samuel Loyd was an American chess player, chess composer, puzzle author, and recreational mathematician. Loyd was born in Philadelphia but raised in New York City.
In visual perception, an optical illusion is an illusion caused by the visual system and characterized by a visual percept that arguably appears to differ from reality. Illusions come in a wide variety; their categorization is difficult because the underlying cause is often not clear but a classification proposed by Richard Gregory is useful as an orientation. According to that, there are three main classes: physical, physiological, and cognitive illusions, and in each class there are four kinds: Ambiguities, distortions, paradoxes, and fictions. A classical example for a physical distortion would be the apparent bending of a stick half immerged in water; an example for a physiological paradox is the motion aftereffect. An example for a physiological fiction is an afterimage. Three typical cognitive distortions are the Ponzo, Poggendorff, and Müller-Lyer illusion. Physical illusions are caused by the physical environment, e.g. by the optical properties of water. Physiological illusions arise in the eye or the visual pathway, e.g. from the effects of excessive stimulation of a specific receptor type. Cognitive visual illusions are the result of unconscious inferences and are perhaps those most widely known.
The tangram is a dissection puzzle consisting of seven flat polygons, called tans, which are put together to form shapes. The objective is to replicate a pattern generally found in a puzzle book using all seven pieces without overlap. Alternatively the tans can be used to create original minimalist designs that are either appreciated for their inherent aesthetic merits or as the basis for challenging others to replicate its outline. It is reputed to have been invented in China sometime around the late 18th century and then carried over to America and Europe by trading ships shortly after. It became very popular in Europe for a time, and then again during World War I. It is one of the most widely recognized dissection puzzles in the world and has been used for various purposes including amusement, art, and education.
An impossible object is a type of optical illusion that consists of a two-dimensional figure which is instantly and naturally understood as representing a projection of a three-dimensional object but cannot exist as a solid object. Impossible objects are of interest to psychologists, mathematicians and artists without falling entirely into any one discipline.
This glossary of chess problems explains commonly used terms in chess problems, in alphabetical order. For a list of unorthodox pieces used in chess problems, see Fairy chess piece; for a list of terms used in chess is general, see Glossary of chess; for a list of chess-related games, see List of chess variants.
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. Each apparently forms a 13×5 right-angled triangle, but one has a 1×1 hole in it.
The Ehrenstein illusion is an optical illusion of brightness or colour perception. The visual phenomena was studied by the German psychologist Walter H. Ehrenstein (1899–1961) who originally wanted to modify the theory behind the Hermann grid illusion. In the discovery of the optical illusion, Ehrenstein found that grating patterns of straight lines that stop at a certain point appear to have a brighter centre, compared to the background.
Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps. Some tiling puzzles ask you to dissect a given shape first and then rearrange the pieces into another shape. Other tiling puzzles ask you to dissect a given shape while fulfilling certain conditions. The two latter types of tiling puzzles are also called dissection puzzles.
A dissection puzzle, also called a transformation puzzle or Richter puzzle, is a tiling puzzle where a set of pieces can be assembled in different ways to produce two or more distinct geometric shapes. The creation of new dissection puzzles is also considered to be a type of dissection puzzle. Puzzles may include various restraints, such as hinged pieces, pieces that can fold, or pieces that can twist. Creators of new dissection puzzles emphasize using a minimum number of pieces, or creating novel situations, such as ensuring that every piece connects to another with a hinge.
A sliding puzzle, sliding block puzzle, or sliding tile puzzle is a combination puzzle that challenges a player to slide pieces along certain routes to establish a certain end-configuration. The pieces to be moved may consist of simple shapes, or they may be imprinted with colours, patterns, sections of a larger picture, numbers, or letters.
In geometry, a dissection problem is the problem of partitioning a geometric figure into smaller pieces that may be rearranged into a new figure of equal content. In this context, the partitioning is called simply a dissection. It is usually required that the dissection use only a finite number of pieces. Additionally, to avoid set-theoretic issues related to the Banach–Tarski paradox and Tarski's circle-squaring problem, the pieces are typically required to be well-behaved. For instance, they may be restricted to being the closures of disjoint open sets.
The T puzzle is a tiling puzzle consisting of four polygonal shapes which can be put together to form a capital T. The four pieces are usually one isosceles right triangle, two right trapezoids and an irregular shaped pentagon.
The nine dots puzzle is a mathematical puzzle whose task is to connect nine squarely arranged points with a pen by four straight lines without lifting the pen.
In ancient Greek geometry, the Ostomachion, also known as loculus Archimedius or syntomachion, is a mathematical treatise attributed to Archimedes. This work has survived fragmentarily in an Arabic version and a copy, the Archimedes Palimpsest, of the original ancient Greek text made in Byzantine times.
In geometry, a square trisection is a type of dissection problem which consists of cutting a square into pieces that can be rearranged to form three identical squares.
In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of Scientific American. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in Mathematics Magazine.
In geometry, a hinged dissection, also known as a swing-hinged dissection or Dudeney dissection, is a kind of geometric dissection in which all of the pieces are connected into a chain by "hinged" points, such that the rearrangement from one figure to another can be carried out by swinging the chain continuously, without severing any of the connections. Typically, it is assumed that the pieces are allowed to overlap in the folding and unfolding process; this is sometimes called the "wobbly-hinged" model of hinged dissection.
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.
The chessboard paradox or paradox of Loyd and Schlömilch 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. Hence the combined area of all four pieces is 64 area units in the square but 65 area units in the rectangle, this seeming contradiction is due an optical illusion as the four pieces don't fit exactly in the rectangle, but leave a small barely visible gap around the rectangle's diagonal. The paradox is sometimes attributed to the American puzzle inventor Sam Loyd (1841–1911) and the German mathematician Oskar Schlömilch (1832–1901)