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Gomory's Theorem
The mutilated chessboard problem is a tiling puzzle posed by Max Black in 1946 that asks: Suppose a standard 8×8 chessboard (or checkerboard) has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2×1 so as to cover all of these squares? It is an impossible puzzle: there is no domino tiling meeting these conditions. One proof of its impossibility uses the fact that, with the corners removed, the chessboard has 32 squares of one color and 30 of the other, but each domino must cover equally many squares of each color. More generally, if any two squares are removed from the chessboard, the rest can be tiled by dominoes if and only if the removed squares are of different colors. This problem has been used as a test case for automated reasoning, creativity, and the philosophy of mathematics. History The mutilated chessboard problem is an instance of domino tiling of grids and polyominoes, also known as "dimer models", a gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutilated Chessboard Vectorized
Mutilation or maiming (from the Latin: ''mutilus'') refers to severe damage to the body that has a ruinous effect on an individual's quality of life. It can also refer to alterations that render something inferior, ugly, dysfunctional, or imperfect. In modern times, the term has an overwhelmingly negative connotation. Terminology In 2019, Michael H. Stone, Gary Brucato, and Ann Burgess proposed formal criteria by which "mutilation" might be systematically distinguished from the act of "dismemberment," as these terms are commonly used interchangeably. They suggested that dismemberment involves "the entire removal, by any means, of a large section of the body of a living or dead person, specifically, the head (also termed decapitation), arms, hands, torso, pelvic area, legs, or feet." Mutilation, by contrast, involves "the removal or irreparable disfigurement, by any means, of some smaller portion of one of those larger sections of a living or dead person. The latter would i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solomon W
Solomon (; , ),, ; ar, سُلَيْمَان, ', , ; el, Σολομών, ; la, Salomon also called Jedidiah ( Hebrew: , Modern: , Tiberian: ''Yăḏīḏăyāh'', "beloved of Yah"), was a monarch of ancient Israel and the son and successor of David, according to the Hebrew Bible and the Old Testament. He is described as having been the penultimate ruler of an amalgamated Israel and Judah. The hypothesized dates of Solomon's reign are 970–931 BCE. After his death, his son and successor Rehoboam would adopt harsh policy towards the northern tribes, eventually leading to the splitting of the Israelites between the Kingdom of Israel in the north and the Kingdom of Judah in the south. Following the split, his patrilineal descendants ruled over Judah alone. The Bible says Solomon built the First Temple in Jerusalem, dedicating the temple to Yahweh, or God in Judaism. Solomon is portrayed as wealthy, wise and powerful, and as one of the 48 Jewish prophets. He is also the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Cycle
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. Determining whether such paths and cycles exist in graphs (the Hamiltonian path problem and Hamiltonian cycle problem) are NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as ''Hamilton's puzzle'', which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ralph E
Ralph (pronounced ; or ,) is a male given name of English, Scottish and Irish origin, derived from the Old English ''Rædwulf'' and Radulf, cognate with the Old Norse ''Raðulfr'' (''rað'' "counsel" and ''ulfr'' "wolf"). The most common forms are: * Ralph, the common variant form in English, which takes either of the given pronunciations. * Rafe, variant form which is less common; this spelling is always pronounced , as are all other English spellings without "l". * Raife, a very rare variant. * Raif, a very rare variant. Raif Rackstraw from H.M.S. Pinafore * Ralf, the traditional variant form in Dutch, German, Swedish, and Polish. * Ralfs, the traditional variant form in Latvian. * Raoul, the traditional variant form in French. * Raúl, the traditional variant form in Spanish. * Raul, the traditional variant form in Portuguese and Italian. * Raül, the traditional variant form in Catalan. * Rádhulbh, the traditional variant form in Irish. Given name Middle Ages * Ralp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gomory's Theorem
The mutilated chessboard problem is a tiling puzzle posed by Max Black in 1946 that asks: Suppose a standard 8×8 chessboard (or checkerboard) has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2×1 so as to cover all of these squares? It is an impossible puzzle: there is no domino tiling meeting these conditions. One proof of its impossibility uses the fact that, with the corners removed, the chessboard has 32 squares of one color and 30 of the other, but each domino must cover equally many squares of each color. More generally, if any two squares are removed from the chessboard, the rest can be tiled by dominoes if and only if the removed squares are of different colors. This problem has been used as a test case for automated reasoning, creativity, and the philosophy of mathematics. History The mutilated chessboard problem is an instance of domino tiling of grids and polyominoes, also known as "dimer models", a gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shmuel Winograd
__NOTOC__ Shmuel Winograd ( he, שמואל וינוגרד; January 4, 1936 – March 25, 2019) was an Israeli-American computer scientist, noted for his contributions to computational complexity. He has proved several major results regarding the computational aspects of arithmetic; his contributions include the Coppersmith–Winograd algorithm and an algorithm for the fast Fourier transform. Winograd studied Electrical Engineering at the Massachusetts Institute of Technology, receiving his B.S. and M.S. degrees in 1959. He received his Ph.D. from the Courant Institute of Mathematical Sciences at New York University in 1968. He joined the research staff at IBM in 1961, eventually becoming director of the Mathematical Sciences Department there from 1970 to 1974 and 1980 to 1994. Honors *IBM Fellow (1972) *Fellow of the Institute of Electrical and Electronics Engineers (1974) * W. Wallace McDowell Award (1974) *Member, National Academy of Sciences (1978) *Member, American Academy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John McCarthy (computer Scientist)
John McCarthy (September 4, 1927 – October 24, 2011) was an American computer scientist and cognitive scientist. He was one of the founders of the discipline of artificial intelligence. He co-authored the document that coined the term "artificial intelligence" (AI), developed the programming language family Lisp, significantly influenced the design of the language ALGOL, popularized time-sharing, and invented garbage collection. McCarthy spent most of his career at Stanford University. He received many accolades and honors, such as the 1971 Turing Award for his contributions to the topic of AI, the United States National Medal of Science, and the Kyoto Prize. Early life and education John McCarthy was born in Boston, Massachusetts, on September 4, 1927, to an Irish immigrant father and a Lithuanian Jewish immigrant mother, John Patrick and Ida (Glatt) McCarthy. The family was obliged to relocate frequently during the Great Depression, until McCarthy's father found work as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Martin Gardner Mathematical Games Columns
Over a period of 24 years (January 1957 – December 1980), Martin Gardner wrote 288 consecutive monthly "Mathematical Games" columns for ''Scientific American'' magazine. During the next years, through June 1986, Gardner wrote 9 more columns, bringing his total to 297, as other authors wrote most of the "Mathematical Games" columns. The table below lists Gardner's columns. Twelve of Gardner's columns provided the cover art for that month's magazine, indicated by "over in the table with a hyperlink to the cover. Other articles by Gardner Gardner wrote 5 other articles for ''Scientific American''. His flexagon article in December 1956 was in all but name the first article in the series of ''Mathematical Games'' columns and led directly to the series which began the following month. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it is the oldest continuously published magazine in the United States. ''Scientific American'' is owned by Springer Nature, which in turn is a subsidiary of Holtzbrinck Publishing Group. History ''Scientific American'' was founded by inventor and publisher Rufus Porter (painter), Rufus Porter in 1845 as a four-page weekly newspaper. The first issue of the large format newspaper was released August 28, 1845. Throughout its early years, much emphasis was placed on reports of what was going on at the United States Patent and Trademark Office, U.S. Patent Office. It also reported on a broad range of inventions including perpetual motion machines, an 1860 device for buoying vessels by Abraham Lincoln, and the universal joint which now can be found ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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