In the history of Europe, the Middle Ages lasted from the 5th to the 15th century. It began with the fall of the Western Roman Empire and merged into the Renaissance and the Age of Discovery; the Middle Ages is the middle period of the three traditional divisions of Western history: classical antiquity, the medieval period, the modern period. The medieval period is itself subdivided into the Early and Late Middle Ages. Population decline, counterurbanisation, collapse of centralized authority and mass migrations of tribes, which had begun in Late Antiquity, continued in the Early Middle Ages; the large-scale movements of the Migration Period, including various Germanic peoples, formed new kingdoms in what remained of the Western Roman Empire. In the 7th century, North Africa and the Middle East—once part of the Byzantine Empire—came under the rule of the Umayyad Caliphate, an Islamic empire, after conquest by Muhammad's successors. Although there were substantial changes in society and political structures, the break with classical antiquity was not complete.
The still-sizeable Byzantine Empire, Rome's direct continuation, survived in the Eastern Mediterranean and remained a major power. The empire's law code, the Corpus Juris Civilis or "Code of Justinian", was rediscovered in Northern Italy in 1070 and became admired in the Middle Ages. In the West, most kingdoms incorporated the few extant Roman institutions. Monasteries were founded; the Franks, under the Carolingian dynasty established the Carolingian Empire during the 8th and early 9th century. It covered much of Western Europe but succumbed to the pressures of internal civil wars combined with external invasions: Vikings from the north, Magyars from the east, Saracens from the south. During the High Middle Ages, which began after 1000, the population of Europe increased as technological and agricultural innovations allowed trade to flourish and the Medieval Warm Period climate change allowed crop yields to increase. Manorialism, the organisation of peasants into villages that owed rent and labour services to the nobles, feudalism, the political structure whereby knights and lower-status nobles owed military service to their overlords in return for the right to rent from lands and manors, were two of the ways society was organised in the High Middle Ages.
The Crusades, first preached in 1095, were military attempts by Western European Christians to regain control of the Holy Land from Muslims. Kings became the heads of centralised nation-states, reducing crime and violence but making the ideal of a unified Christendom more distant. Intellectual life was marked by scholasticism, a philosophy that emphasised joining faith to reason, by the founding of universities; the theology of Thomas Aquinas, the paintings of Giotto, the poetry of Dante and Chaucer, the travels of Marco Polo, the Gothic architecture of cathedrals such as Chartres are among the outstanding achievements toward the end of this period and into the Late Middle Ages. The Late Middle Ages was marked by difficulties and calamities including famine and war, which diminished the population of Europe. Controversy and the Western Schism within the Catholic Church paralleled the interstate conflict, civil strife, peasant revolts that occurred in the kingdoms. Cultural and technological developments transformed European society, concluding the Late Middle Ages and beginning the early modern period.
The Middle Ages is one of the three major periods in the most enduring scheme for analysing European history: classical civilisation, or Antiquity. The "Middle Ages" first appears in Latin in 1469 as media tempestas or "middle season". In early usage, there were many variants, including medium aevum, or "middle age", first recorded in 1604, media saecula, or "middle ages", first recorded in 1625; the alternative term "medieval" derives from medium aevum. Medieval writers divided history into periods such as the "Six Ages" or the "Four Empires", considered their time to be the last before the end of the world; when referring to their own times, they spoke of them as being "modern". In the 1330s, the humanist and poet Petrarch referred to pre-Christian times as antiqua and to the Christian period as nova. Leonardo Bruni was the first historian to use tripartite periodisation in his History of the Florentine People, with a middle period "between the fall of the Roman Empire and the revival of city life sometime in late eleventh and twelfth centuries".
Tripartite periodisation became standard after the 17th-century German historian Christoph Cellarius divided history into three periods: ancient and modern. The most given starting point for the Middle Ages is around 500, with the date of 476 first used by Bruni. Starting dates are sometimes used in the outer parts of Europe. For Europe as a whole, 1500 is considered to be the end of the Middle Ages, but there is no universally agreed upon end date. Depending on the context, events such as the conquest of Constantinople by the Turks in 1453, Christopher Columbus's first voyage to the Americas in 1492, or the Protestant Reformation in 1517 are sometimes used. English historians use the Battle of Bosworth Field in 1485 to mark the end of the period. For Spain, dates used are the death of King Ferdinand II in 1516, the death of Queen Isabella I of Castile in 1504, or the conquest of Granada in 1492. Historians from Romance-speaking countries tend to divide the Middle Ages into two parts: an earlier "High" and late
Leonhard Euler was a Swiss mathematician, astronomer and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while making pioneering contributions to several branches such as topology and analytic number theory. He introduced much of the modern mathematical terminology and notation for mathematical analysis, such as the notion of a mathematical function, he is known for his work in mechanics, fluid dynamics, optics and music theory. Euler was one of the most eminent mathematicians of the 18th century and is held to be one of the greatest in history, he is widely considered to be the most prolific mathematician of all time. His collected works fill more than anybody in the field, he spent most of his adult life in Saint Petersburg, in Berlin the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, Marguerite née Brucker, a pastor's daughter.
He had two younger sisters: Anna Maria and Maria Magdalena, a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Paul Euler was a friend of the Bernoulli family. Euler's formal education started in Basel. In 1720, aged thirteen, he enrolled at the University of Basel, in 1723, he received a Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who discovered his new pupil's incredible talent for mathematics. At that time Euler's main studies included theology and Hebrew at his father's urging in order to become a pastor, but Bernoulli convinced his father that Leonhard was destined to become a great mathematician. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono. At that time, he was unsuccessfully attempting to obtain a position at the University of Basel.
In 1727, he first entered the Paris Academy Prize Problem competition. Pierre Bouguer, who became known as "the father of naval architecture", won and Euler took second place. Euler won this annual prize twelve times. Around this time Johann Bernoulli's two sons and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia, when Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727, he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration.
Euler settled into life in Saint Petersburg. He took on an additional job as a medic in the Russian Navy; the Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made attractive to foreign scholars like Euler; the academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Few students were enrolled in the academy in order to lessen the faculty's teaching burden, the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions; the Academy's benefactress, Catherine I, who had continued the progressive policies of her late husband, died on the day of Euler's arrival. The Russian nobility gained power upon the ascension of the twelve-year-old Peter II; the nobility was suspicious of the academy's foreign scientists, thus cut funding and caused other difficulties for Euler and his colleagues.
Conditions improved after the death of Peter II, Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731. Two years Daniel Bernoulli, fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department. On 7 January 1734, he married Katharina Gsell, a daughter of Georg Gsell, a painter from the Academy Gymnasium; the young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood. Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia, he lived for 25 years in Berlin. In Berlin, he published the two works for which he would become most renowned: the Introductio in analysin infinitorum, a text on functions published in 1748, the Institutiones calculi differentialis, published in 1755 on differential calculus.
A puzzle is a game, problem, or toy that tests a person's ingenuity or knowledge. In a puzzle, the solver is expected to put pieces together in a logical way, in order to arrive at the correct or fun solution of the puzzle. There are different genres of puzzles, such as crossword puzzles, word-search puzzles, number puzzles, relational puzzles, or logic puzzles. Puzzles are created to be a form of entertainment but they can arise from serious mathematical or logistical problems. In such cases, their solution may be a significant contribution to mathematical research; the 1989 edition of the Oxford English Dictionary dates the word puzzle to the end of the 16th century. Its first documented use was in a book titled The Voyage of Robert Dudley...to the West Indies, 1594–95, narrated by Capt. Wyatt, by himself, by Abram Kendall, master; the word came to be used as a noun. The word puzzle comes from pusle, meaning "bewilder, confound", a frequentive of the obsolete verb pose in the sense of "perplex".
The use of the word to mean "a toy contrived to test one's ingenuity" is recent. Puzzles can be divided into categories. For example, a maze is a type of tour puzzle; some other categories are construction puzzles, stick puzzles, tiling puzzles, disentanglement puzzles, lock puzzles, folding puzzles, combination puzzles, mechanical puzzles. A chess problem is a puzzle. Examples are the eight queens puzzle. Jigsaw puzzles. Lateral thinking puzzles called "situation puzzles" Mathematical puzzles include the missing square puzzle and many impossible puzzles — puzzles which have no solution, such as the Seven Bridges of Königsberg, the three cups problem, three utilities problem Mechanical puzzles such as the Rubik's Cube and Soma cube Metapuzzles are puzzles which unite elements of other puzzles. Paper-and-pencil puzzles such as Uncle Art's Funland, connect the dots, nonograms Also the logic puzzles published by Nikoli: Sudoku, Kakuro, Hashiwokakero, Hitori, Light Up, Number Link, Ripple Effect and Kuromasu.
Peg solitaire. Rubik's Cube and other combination puzzles can be stimulating toys for children or recreational activities for adults. Sangaku Sliding puzzles such as the 15 Puzzle. Puzz-3D is a three-dimensional variant of this type. Sokoban Spot the difference Tangram Word puzzles, including anagrams, crossword puzzles and word search puzzles. Tabletop and digital word puzzles include Bananagrams, Bonza, Letterpress, Puzzlage, Ruzzle, Upwords, WordSpot, Words with Friends. Wheel of Fortune is a game show centered on a word puzzle. Solutions of puzzles require the recognition of patterns and the adherence to a particular kind of ordering. People with a high level of inductive reasoning aptitude may be better at solving such puzzles than others, but puzzles based upon inquiry and discovery may be solved more by those with good deduction skills. Deductive reasoning improves with practice. Mathematical puzzles involves BODMAS. BODMAS is an acronym and it stands for Bracket, Of, Multiplication and Subtraction.
In certain regions, PEDMAS is the synonym of BODMAS. It explains the order of operations to solve an expression; some mathematical puzzle requires Top to Bottom convention to avoid the ambiguity in the order of operations. It is an elegantly simple idea that relies, as sudoku does, on the requirement that numbers appear only once starting from top to bottom as coming along. Puzzle makers are people; some notable creators of puzzles are: Ernő Rubik Sam Loyd Henry Dudeney Boris Kordemsky David J. Bodycombe Will Shortz Lloyd King Martin Gardner Raymond Smullyan Jigsaw puzzles are the most popular form of puzzle. Jigsaw puzzles were invented around 1760, when John Spilsbury, a British engraver and cartographer, mounted a map on a sheet of wood, which he sawed around the outline of each individual country on the map, he used the resulting pieces as an aid for the teaching of geography. After becoming popular among the public, this kind of teaching aid remained the primary use of jigsaw puzzles until about 1820.
The largest puzzle is made by German game company Ravensburger. The smallest puzzle made was created at LaserZentrum Hannover, it is the size of a sand grain. By the early 20th century and newspapers had found that they could increase their readership by publishing puzzle contests, beginning with crosswords and in modern days sudoku. There are organizations and events that cater to puzzle enthusiasts, such as: Nob Yoshigahara Puzzle Design Competition World Puzzle Championship National Puzzlers' League Puzzlehunts such as the Maze of Games List of impossible puzzles List of Nikoli puzzle types Riddle Puzzles at DMOZ
Checkmate is a game position in chess and other chess-like games in which a player's king is in check and there is no way to remove the threat. Checkmating the opponent wins the game. In chess, the king is never captured—the game ends as soon as the king is checkmated. In formal games, most players resign an lost game before being checkmated, it is considered bad etiquette to continue playing in a hopeless position. If a player is not in check but has no legal move it is stalemate, the game ends in a draw. A checkmating move is recorded in algebraic notation using the hash symbol "#", for example: 34. Qg7#. A checkmate may occur in as few as two moves on one side with all of the pieces still on the board, in a middlegame position, or after many moves with as few as three pieces in an endgame position; the term checkmate is, according to the Barnhart Etymological Dictionary, an alteration of the Persian phrase "shāh māt" which means "the King is helpless". Persian "māt" applies to the king but in Sanskrit "māta" pronounced "māt", applied to his kingdom "traversed, measured across, meted out" by his opponent.
Others maintain that it means "the King is dead", as chess reached Europe via the Islamic world, Arabic māta means "died" or "is dead". However, in Pashto, the word māt still exists, meaning "destroyed, broken". Moghadam traced the etymology of the word mate, it comes from a Persian verb mandan, meaning "to remain", cognate with the Latin word maneō and the Greek menō. It means "remained" in the sense of "abandoned" and the formal translation is "surprised", in the military sense of "ambushed". Sheikh is the Arabic word for the monarch. Players would announce "Sheikh". "Māt" is an Arabic adjective for dead "helpless", or "defeated". So the king is in mate when he is ambushed, at a loss, defeated, or abandoned to his fate. In modern Arabic, the word mate depicts a person who has died open-mouthed, staring and unresponsive; the words "stupefied" or "stunned" bear close correlation. So a possible alternative would be to interpret mate as "unable to respond". A king is mate means a king is unable to respond, which would correspond to there being no response that a player's king can make to their opponent's final move.
This interpretation is much closer to the original intent of the game being not to kill a king but to leave him with no viable response other than surrender, which better matches the origin story detailed in the Shahnameh. In modern parlance, the term checkmate is a metaphor for an strategic victory. In early Sanskrit chess, the king could be captured and this ended the game; the Persians introduced the idea of warning. This was done to avoid the accidental end of a game; the Persians added the additional rule that a king could not be moved into check or left in check. As a result, the king could not be captured, checkmate was the only decisive way of ending a game. Before about 1600, the game could be won by capturing all of the opponent's pieces, leaving just a bare king; this style of play is now called robado. In Medieval times, players began to consider it nobler to win by checkmate, so annihilation became a half-win for a while, until it was abandoned. Two major pieces can force checkmate on the edge of the board.
The process is to put the two pieces on adjacent ranks or files and force the king to the side of the board, where one piece keeps the king on the edge of the board while the other delivers checkmate. In the illustration, white checkmates by forcing the black king to one row at a time; the same process can be used to checkmate with two rooks, or with two queens. There are four fundamental checkmates when one side has only his king and the other side has only the minimum material needed to force checkmate, i.e. one queen, one rook, two bishops on opposite-colored squares, or a bishop and a knight. The king must help in accomplishing all of these checkmates. If the superior side has more material, checkmates are easier; the checkmate with the queen is the most common, easiest to achieve. It occurs after a pawn has queened. A checkmate with the rook is common, but a checkmate with the two bishops or with a bishop and knight occurs infrequently; the two bishop checkmate is easy to accomplish, but the bishop and knight checkmate is difficult and requires precision.
The first two diagrams show representatives of the basic checkmate positions with a queen, which can occur on any edge of the board. The exact position can vary from the diagram. In the first of the checkmate positions, the queen is directly in front of the opposing king and the white king is protecting its queen. In the second checkmate position, the kings are in opposition and the queen mates on the rank of the king. With the side with the queen to move, checkmate can be forced in at most ten moves from any starting position, with optimal play by both sides, but fewer moves are required. In positions in which a pawn has just promoted to a queen, at most nine moves are required. In the position diagrammed, White checkmates by confining the black king to a rectangle and shrinking the rectangle to force the king to the edge
Wheat and chessboard problem
The wheat and chessboard problem is a mathematical problem expressed in textual form as: If a chessboard were to have wheat placed upon each square such that one grain were placed on the first square, two on the second, four on the third, so on, how many grains of wheat would be on the chessboard at the finish? The problem may be solved using simple addition. With 64 squares on a chessboard, if the number of grains doubles on successive squares the sum of grains on all 64 squares is: 1 + 2 + 4 + 8 +... and so forth for the 64 squares. The total number of grains equals 18,446,744,073,709,551,615, much higher; this exercise can be used to demonstrate how exponential sequences grow, as well as to introduce exponents, zero power, capital-sigma notation and geometric series. Updated for modern times using pennies and the hypothetical question, "Would you rather have a million dollars or the sum of a penny doubled every day for a month?", the formula has been used to explain compounded interest. The problem appears in different stories about the invention of chess.
One of them includes the geometric progression problem. The story is first known to have been recorded in 1256 by Ibn Khallikan. Another version has the inventor of chess request his ruler give him wheat according to the wheat and chessboard problem; the ruler laughs it off as a meager prize for a brilliant invention, only to have court treasurers report the unexpectedly huge number of wheat grains would outstrip the ruler's resources. Versions differ as to whether the inventor is executed. Macdonnell investigates the earlier development of the theme. Shatranj, or chess was invented under an Indian king, who expressed his preference for this game over backgammon; the Indians, he adds calculated an arithmetical progression with the squares of the chessboard. The early fondness of the Indians for enormous calculations is well known to students of their mathematics, is exemplified in the writings of the great astronomer Āryabaṭha. An additional argument for the Indian origin of this calculation is supplied by the Arabic name for the square of the chessboard,'house'.
For this has doubtless a historical connection with its Indian designation koṣṭhāgāra,'store-house','granary'. The simple, brute-force solution is just to manually double and add each step of the series: T 64 = 1 + 2 + 4 + ⋯ + 9, 223, 372, 036, 854, 775, 808 = 18, 446, 744, 073, 709, 551, 615 where T 64 is the total number of grains; the series may be expressed using exponents: T 64 = 2 0 + 2 1 + 2 2 + ⋯ + 2 63 and, represented with capital-sigma notation as: ∑ i = 0 63 2 i. It can be solved much more using: T 64 = 2 64 − 1. A proof of which is: s = 2 0 + 2 1 + 2 2 + ⋯ + 2 63. Multiply each side by 2: 2 s = 2 1 + 2 2 + 2 3 + ⋯ + 2 63 + 2 64. Subtract original series from each side: 2 s − s = 2 64 − 2 0 ∴ s = 2 64 − 1; the solution above is a particular case of the sum of a geometric series, given by a + a r + a r 2 + a r 3 + ⋯ + a r n − 1 = ∑ k = 0 n − 1 a r k = a 1 − r n 1 − r, where a is the first term of the series, r is the common ratio and n is the number of terms. In this problem a = 1, r = 2
Adrien-Marie Legendre was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him. Adrien-Marie Legendre was born in Paris on 18 September 1752 to a wealthy family, he received his education at the Collège Mazarin in Paris, defended his thesis in physics and mathematics in 1770. He taught at the École Militaire in Paris from 1775 to 1780 and at the École Normale from 1795. At the same time, he was associated with the Bureau des Longitudes. In 1782, the Berlin Academy awarded Legendre a prize for his treatise on projectiles in resistant media; this treatise brought him to the attention of Lagrange. The Académie des sciences made Legendre an adjoint member in 1783 and an associate in 1785. In 1789, he was elected a Fellow of the Royal Society, he assisted with the Anglo-French Survey to calculate the precise distance between the Paris Observatory and the Royal Greenwich Observatory by means of trigonometry.
To this end in 1787 he visited Dover and London together with Dominique, comte de Cassini and Pierre Méchain. The three visited William Herschel, the discoverer of the planet Uranus. Legendre lost his private fortune in 1793 during the French Revolution; that year, he married Marguerite-Claudine Couhin, who helped him put his affairs in order. In 1795, Legendre became one of six members of the mathematics section of the reconstituted Académie des Sciences, renamed the Institut National des Sciences et des Arts. In 1803, Napoleon reorganized the Institut National, Legendre became a member of the Geometry section. From 1799 to 1812, Legendre served as mathematics examiner for graduating artillery students at the École Militaire and from 1799 to 1815 he served as permanent mathematics examiner for the École Polytechnique. In 1824, Legendre's pension from the École Militaire was stopped because he refused to vote for the government candidate at the Institut National, his pension was reinstated with the change in government in 1828.
In 1831, he was made an officer of the Légion d'Honneur. Legendre died in Paris on 10 January 1833, after a long and painful illness, Legendre's widow preserved his belongings to memorialize him. Upon her death in 1856, she was buried next to her husband in the village of Auteuil, where the couple had lived, left their last country house to the village. Legendre's name is one of the 72 names inscribed on the Eiffel Tower. Abel's work on elliptic functions was built on Legendre's and some of Gauss' work in statistics and number theory completed that of Legendre, he developed the least squares method and firstly communicated it to his contemporaries before Gauss, which has broad application in linear regression, signal processing and curve fitting. Today, the term "least squares method" is used as a direct translation from the French "méthode des moindres carrés", his major work is Exercices de Calcul Intégral, published in three volumes in 1811, 1817 and 1819. In the first volume he introduced the basic properties of elliptic integrals, beta functions and gamma functions, introducing the symbol Γ normalizing it to Γ = n!.
Further results on the beta and gamma functions along with their applications to mechanics - such as the rotation of the earth, the attraction of ellipsoids, appeared in the second volume. In 1830, he gave a proof of Fermat's last theorem for exponent n = 5, proven by Lejeune Dirichlet in 1828. In number theory, he conjectured the quadratic reciprocity law, subsequently proved by Gauss, he did pioneering work on the distribution of primes, on the application of analysis to number theory. His 1798 conjecture of the prime number theorem was rigorously proved by Hadamard and de la Vallée-Poussin in 1896. Legendre did an impressive amount of work on elliptic functions, including the classification of elliptic integrals, but it took Abel's stroke of genius to study the inverses of Jacobi's functions and solve the problem completely, he is known for the Legendre transformation, used to go from the Lagrangian to the Hamiltonian formulation of classical mechanics. In thermodynamics it is used to obtain the enthalpy and the Helmholtz and Gibbs energies from the internal energy.
He is the namesake of the Legendre polynomials, solutions to Legendre's differential equation, which occur in physics and engineering applications, e.g. electrostatics. Legendre is best known as the author of Éléments de géométrie, published in 1794 and was the leading elementary text on the topic for around 100 years; this text rearranged and simplified many of the propositions from Euclid's Elements to create a more effective textbook. Foreign Honorary Member of the American Academy of Arts and Sciences The Moon crater Legendre is named after him. Main-belt asteroid. Legendre is one of the 72 prominent French scientists who were commemorated on plaques at the first stage of the Eiffel Tower when it first opened. Essays1782 Recherches sur la trajectoire des projectiles dans les milieux résistants BooksEléments de géométrie, textbook 1794 Essai sur la Théorie des Nombres 1797-8, 2nd ed. 1808, 3rd ed. in 2 vol. 1830 Nouvelles Méthodes pour la Détermination des Orbites des Comètes, 1805 Exercices de Calcul Intégral, book in three volumes 1811, 1817, 1819 Traité des Fonctions Elliptiques, book in three volumes 1825, 1826, 1830Memoires in Histoire de l'Académie Royale des Scien
Mutilated chessboard problem
The mutilated chessboard problem is a tiling puzzle proposed by philosopher Max Black in his book Critical Thinking. It was discussed by Solomon W. Golomb, Gamow & Stern and by Martin Gardner in his Scientific American column "Mathematical Games"; the problem is as follows: Suppose a standard 8×8 chessboard 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? Most considerations of this problem in literature provide solutions "in the conceptual sense" without proofs. John McCarthy proposed it as a hard problem for automated proof systems. In fact, its solution using the resolution system of inference is exponentially hard; the puzzle is impossible to complete. A domino placed on the chessboard will always cover one black square. Therefore, a collection of dominoes placed on the board will cover an equal numbers of squares of each color. If the two white corners are removed from the board 30 white squares and 32 black squares remain to be covered by dominoes, so this is impossible.
If the two black corners are removed instead 32 white squares and 30 black squares remain, so it is again impossible. The same impossibility proof shows that no domino tiling exists whenever any two white squares are removed from the chessboard. However, if two squares of opposite colors are removed it is always possible to tile the remaining board with dominoes. Gomory's theorem can be proven using a Hamiltonian cycle of the grid graph formed by the chessboard squares. Tiling a rectangle with tetrominoes Gamow, George.