À esquerda, um quadrado geomágico 3x3 onde cada elemento é um decômino – um polinômio feito de dez unidades quadradas. Os decôminos em cada fileira, coluna e diagonal podem ser arranjados juntos para formar um retângulo 5x6. (Eles podem ser montados de duas maneiras diferentes, que são desenhadas em qualquer lado de cada fileira, coluna e diagonal.) Este é apenas um de dois quadrados geomágicos possíveis quando as peças são todas decôminos - tornando esse tipo de quadrado geomágico muito mais raro do que quando todos os pentôminos de um a nove são usados. À direita, um quadrado geomágico 3x3 one cada elemento é um polinômio, variando em tamanho entre um e nove. Os quadrados em torno da periferia mostram como os polinômios das fileiras, colunas e diagonais correspondentes podem ser arranjados juntos para formar a mesma figura, nesse caso, um quadrado 4x4 com uma célula interna removida. Sallows descobriu 4.370 quadrados como esse, consistindo nos polinômios de um a nove, e fazendo a mesma forma alvo - muitos mais do que ele esperava.
Um quadrado mágico é uma grade de números para os quais cada linha, coluna e diagonais somam o mesmo número. Por exemplo:
4 9 2
3 5 7
8 1 6
Neste caso, as linhas, colunas e diagonais todos iguais a 15.
O quadrado mágico é um conceito simples e que tem sido em torno de milhares de anos. Matemáticos, artistas e místicos têm sido fascinados pelos padrões hipnotizantes que eles produzem, e talvez não seja surpreendente que Sallows, sob a influência de drogas alucinógenas, foi tomado por sua magia.
Depois que o efeito do ácido cessou, porém, o interesse Sallows por quadrados mágicos não diminuiu. Tornou-se uma obsessão. Na verdade, ele se tornou tão fixada neles que, depois de um par de décadas de contemplação, ele descobriu uma nova categoria de quadrado, que está abrindo um capítulo novo, fascinante, para um dos mais antigos gêneros da matemática.
Sallows fez uma coisa muito simples, embora nunca tivesse sido feita antes. Ele encheu seus quadrados com formas geométricas em vez de números. Em vez de cada linha, coluna ou diagonal a somar o mesmo número, as formas em cada linha, coluna ou diagonal tiveram que se encaixar para criar a mesma forma-mestre.
Ele chamou sua invenção do "Quadrado Geomagic". Eles são uma maneira inteiramente nova de apreciar as sutilezas de quadrados mágicos, e proporcionar belos padrões geométricos, também.
Talvez nenhuma outra área da matemática não-prática tenha sido tão popular por tanto tempo como quadrados mágicos. Sua história começa há 4.000 anos na China, onde, segundo a lenda, uma tartaruga se arrastou para fora do Rio Amarelo. O réptil é conhecido por ter pontos em sua parte inferior posicionados de tal forma a fornar o quadrado 3x3 descrito acima.
Os chineses chamaram-no quadrado shu lo, e deram-lhe importância espiritual, acreditando que ele encapsulava as harmonias do universo. Feng shui, o sistema chinês de organizar objetos, como móveis em residências, é em parte baseada na shu lo.
Mas a veneração pelos quadrados mágicos não se limitou aos chineses. Na Índia, quadrados mágicos eram usados como amuletos de proteção a virgens. Na Turquia eram bordados quadrados mágicos nas camisas dos guerreiros e na Europa Ocidental do Renascimento astrólogos os relacionavam a planetas.
Em 1514, o artista e matemático alemão Albrecht Dürer produziu sua obra mais famosa, Melencolia I, uma xilogravura representando um anjo em apuros cercado por objetos científicos, que incluiam o quadrado 4x4 seguinte:
16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1
Este quadrado é particularmente surpreendente. Não só as linhas - linhas e diagonais somam 34 -, mas os quatro cantos, os quatro dígitos no quadrado central, e os quatro dígitos no canto superior esquerdo, superior direito, inferior esquerdo e quartos inferiores direitos também. Há muitas outras combinações de quatro números no quadrado que se somam aos 34 anos, e é divertido olhar para eles. Dürer ainda incluiu o ano em que ele fez a gravação - 1514 - na linha de fundo.
Alguns grandes matemáticos estudaram quadrados mágicos - como Leonhard Euler no século 18, e Lucas Édouard e Arthur Cayley no século 19 - mas o campo tem sido geralmente domínio de leigos apaixonados. O mais notável foi aficionado, nos Estados Unidos, foi Benjamin Franklin, que gostava de passar seu tempo livre especialmente na construção de variantes inovadoras. Em uma noite de seus 40 anos ele compôs um quadrado 16x16 que ele dizia ser "o mais mágico passe de mágica de qualquer quadrado mágico já feito por qualquer mago".
Sallows, 66 anos, é um amador muito entusiasta. Ele deixou a escola aos 17 anos sem qualificação; é totalmente autodidata em matemática . Era um engenheiro eletrônico da equipe não docente da Universidade de Nijmegen, até que se aposentou há dois anos.
Enquantoé certamente notável que um matemático trouxe a este campo um novo sopro de vida, ao contrário do que seria de se esperar, a maioria dos matemáticos acadêmicos passou a considerar quadrados mágicos como demasiado frívolos para ocupar seu tempo.
O quadrado mágico convencional tem n linhas e n colunas, e deve incluir todos os números de 1 a n ². Assim, como acima, o 3x3 tem todos os números de um a nove e os 4x4 número a cada um a 16.
Sallows diz que ele foi imediatamente "acionado pelas simetrias" dos quadrados mágicos, e uma vez que ele se "contaminou", começou a inventar novas regras e modificações. Ele é enaltecido ter inventado o "quadrado alphamagic", que é duplamente mágico:
cinco 22 dezoito
2008 1502
1208 2005
As linhas, colunas e diagonais somam 45 ao considerar-se o significado das palavras. Mas quando se considera o número de letras de cada palavra - para cinco anos é 4 e 22 é 9 - este também gera um quadrado mágico, cujas linhas, colunas e diagonais somam 21. Outra propriedade fantástica dele são os comprimentos das palavras compostas pelos números consecutivos 3-11.
Sallows ganhou certa fama no mundo da matemática recreativa para seu quadrado alphamagic, mas foi considerado um "bobo, um bocado parvo".
Ele estava mais animado, enxergando por trás a álgebra nos quadrados mágicos. Édouard Lucas tinha descoberto esta fórmula, o que gera um quadrado mágico quando você substitui a, b e c para os números:
a + b a-b-c a + c
a-b + c a a + b-c
A-C a + b + c a-b
Quando você deixa um b = 5, = -1 e c = -3, você obtém o Eis Shu.
Sallows estava brincando com a fórmula de Lucas: "E então ele veio a mim - porque eu não represento as variáveis por formas um 'mais' poderia ser acrescentando, e uma extirpação de um 'menos'? Daí se seguiu todo o resto. É absolutamente incrível que ninguém havia pensado nisso antes. Eu soube imediatamente que tinha tido uma idéia muito importante. Minha mente estava cheia de possibilidades. "
Como vimos acima, as linhas, colunas e diagonais de um quadrado mágico somam o mesmo número. Se cada número no quadrado é representado por uma linha de tal extensão, segue-se que estes segmentos de linha podem ser unidos de um extremo a outro para formar uma linha maior - e essa linha tem o mesmo comprimento de qualquer linha, coluna ou diagonal que você escolher .
Da mesma forma, se cada número n é substituído por uma forma que n tem espaço, segue-se que as formas em cada linha, coluna ou diagonal, quando colocadas juntas têm a mesma área combinada.
Sallows ainda queria a condição extra de que as formas se encaixam de forma que cada linha, coluna ou diagonal toma exatamente a mesma forma de destino.
"Foi um negócio muito complicado", disse ele. Usando um computador demorei um longo tempo para encontrar formas que trabalhei e quando ele o fiz: "Eu realmente caí da cadeira."
Sallows tinha a impressão de que quadrados Geomagic seriam poucos e distantes entre si. No entanto, ele encontrou milhares deles. "Foi o oposto completo do que eu esperava. Eu estava procurando por algo incrivelmente raro, mas descobri que não era nada raro."
O Eis Shu é o quadrado mágico de 3x3 só que é possível construir quando os números de um a nove são todos usados. (Excluindo rodá-los e refleti-los, que são considerados matematicamente equivalentes.)
Mas quando o Sallows encheu seu quadrado 3x3 com nove Geomagic polnômios que têm áreas de 1-9 - isto é, formas feitas a partir de um quadrado unitário, dois quadrados da unidade, três, e assim sucessivamente até nove quadrados unidade - ele descobriu que existem 1.411quadrados Geomagic , quando a forma-alvo é um retângulo 3x5. Ele foi capaz de encontrar 4.370 quadrados Geomagic onde a forma-alvo é um quadrado de 4x4 faltando uma peça central, 27.110, quando o alvo é um quadrado de 4x4 faltando uma peça de canto e 16.465 quando ao alvo está faltando uma peça não de canto.
Sallows reivindica que suas descobertas lançaram novas luzes sobre os quadrados mágicos. "Ninguém entendeu o que é um quadrado mágico", diz ele. "Nós sempre assumos que são objetos numéricos. A partir dessa perspectiva de que são todos os objectos geométricos, apenas um pequeno subconjunto são objetos numéricos."
Sallows passou a encontrar dezenas de novos tipos de quadrados Geomagic , envolvendo associações de todos os tipos de formas. Ele fez uma galeria de suas descobertas na web no Natal do ano passado, e desde então mais de 34.000 pessoas de 129 países visitaram seu site - tornando-se um hit viral no mundo da matemática recreativa.
Poderia Sallows ver neles alguma utilidade? "Eu não acredito que este material seja da menor importância", responde ele. "É agradável e charmoso e agrada a todos, mas não tem aplicação alguma."
Peter Cameron, da Universidade de Londres, é mais prudente, uma vez que as aplicações são difíceis de prever: ". Euler descobriu quadrados latinos, quando ele estava fazendo uma nova construção de quadrados mágicos, estes agora têm ampla aplicação em muitas áreas, incluindo delineamento experimental. Talvez haja "quadrados geo-Latinos", que tenham alguma aplicação. "
Sejam ou não encontrados usos, contudo, Sallows certamente provou uma coisa: que o fascínio milenar por quadrados mágicos ainda não acabou.
Alex Bellos é o autor das aventuras de Alex em Numberland (Terra dos Números)
4 9 2
3 5 7
8 1 6
Neste caso, as linhas, colunas e diagonais todos iguais a 15.
O quadrado mágico é um conceito simples e que tem sido em torno de milhares de anos. Matemáticos, artistas e místicos têm sido fascinados pelos padrões hipnotizantes que eles produzem, e talvez não seja surpreendente que Sallows, sob a influência de drogas alucinógenas, foi tomado por sua magia.
Depois que o efeito do ácido cessou, porém, o interesse Sallows por quadrados mágicos não diminuiu. Tornou-se uma obsessão. Na verdade, ele se tornou tão fixada neles que, depois de um par de décadas de contemplação, ele descobriu uma nova categoria de quadrado, que está abrindo um capítulo novo, fascinante, para um dos mais antigos gêneros da matemática.
Sallows fez uma coisa muito simples, embora nunca tivesse sido feita antes. Ele encheu seus quadrados com formas geométricas em vez de números. Em vez de cada linha, coluna ou diagonal a somar o mesmo número, as formas em cada linha, coluna ou diagonal tiveram que se encaixar para criar a mesma forma-mestre.
Ele chamou sua invenção do "Quadrado Geomagic". Eles são uma maneira inteiramente nova de apreciar as sutilezas de quadrados mágicos, e proporcionar belos padrões geométricos, também.
Talvez nenhuma outra área da matemática não-prática tenha sido tão popular por tanto tempo como quadrados mágicos. Sua história começa há 4.000 anos na China, onde, segundo a lenda, uma tartaruga se arrastou para fora do Rio Amarelo. O réptil é conhecido por ter pontos em sua parte inferior posicionados de tal forma a fornar o quadrado 3x3 descrito acima.
Os chineses chamaram-no quadrado shu lo, e deram-lhe importância espiritual, acreditando que ele encapsulava as harmonias do universo. Feng shui, o sistema chinês de organizar objetos, como móveis em residências, é em parte baseada na shu lo.
Mas a veneração pelos quadrados mágicos não se limitou aos chineses. Na Índia, quadrados mágicos eram usados como amuletos de proteção a virgens. Na Turquia eram bordados quadrados mágicos nas camisas dos guerreiros e na Europa Ocidental do Renascimento astrólogos os relacionavam a planetas.
Em 1514, o artista e matemático alemão Albrecht Dürer produziu sua obra mais famosa, Melencolia I, uma xilogravura representando um anjo em apuros cercado por objetos científicos, que incluiam o quadrado 4x4 seguinte:
16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1
Este quadrado é particularmente surpreendente. Não só as linhas - linhas e diagonais somam 34 -, mas os quatro cantos, os quatro dígitos no quadrado central, e os quatro dígitos no canto superior esquerdo, superior direito, inferior esquerdo e quartos inferiores direitos também. Há muitas outras combinações de quatro números no quadrado que se somam aos 34 anos, e é divertido olhar para eles. Dürer ainda incluiu o ano em que ele fez a gravação - 1514 - na linha de fundo.
Alguns grandes matemáticos estudaram quadrados mágicos - como Leonhard Euler no século 18, e Lucas Édouard e Arthur Cayley no século 19 - mas o campo tem sido geralmente domínio de leigos apaixonados. O mais notável foi aficionado, nos Estados Unidos, foi Benjamin Franklin, que gostava de passar seu tempo livre especialmente na construção de variantes inovadoras. Em uma noite de seus 40 anos ele compôs um quadrado 16x16 que ele dizia ser "o mais mágico passe de mágica de qualquer quadrado mágico já feito por qualquer mago".
Sallows, 66 anos, é um amador muito entusiasta. Ele deixou a escola aos 17 anos sem qualificação; é totalmente autodidata em matemática . Era um engenheiro eletrônico da equipe não docente da Universidade de Nijmegen, até que se aposentou há dois anos.
Enquantoé certamente notável que um matemático trouxe a este campo um novo sopro de vida, ao contrário do que seria de se esperar, a maioria dos matemáticos acadêmicos passou a considerar quadrados mágicos como demasiado frívolos para ocupar seu tempo.
O quadrado mágico convencional tem n linhas e n colunas, e deve incluir todos os números de 1 a n ². Assim, como acima, o 3x3 tem todos os números de um a nove e os 4x4 número a cada um a 16.
Sallows diz que ele foi imediatamente "acionado pelas simetrias" dos quadrados mágicos, e uma vez que ele se "contaminou", começou a inventar novas regras e modificações. Ele é enaltecido ter inventado o "quadrado alphamagic", que é duplamente mágico:
cinco 22 dezoito
2008 1502
1208 2005
As linhas, colunas e diagonais somam 45 ao considerar-se o significado das palavras. Mas quando se considera o número de letras de cada palavra - para cinco anos é 4 e 22 é 9 - este também gera um quadrado mágico, cujas linhas, colunas e diagonais somam 21. Outra propriedade fantástica dele são os comprimentos das palavras compostas pelos números consecutivos 3-11.
Sallows ganhou certa fama no mundo da matemática recreativa para seu quadrado alphamagic, mas foi considerado um "bobo, um bocado parvo".
Ele estava mais animado, enxergando por trás a álgebra nos quadrados mágicos. Édouard Lucas tinha descoberto esta fórmula, o que gera um quadrado mágico quando você substitui a, b e c para os números:
a + b a-b-c a + c
a-b + c a a + b-c
A-C a + b + c a-b
Quando você deixa um b = 5, = -1 e c = -3, você obtém o Eis Shu.
Sallows estava brincando com a fórmula de Lucas: "E então ele veio a mim - porque eu não represento as variáveis por formas um 'mais' poderia ser acrescentando, e uma extirpação de um 'menos'? Daí se seguiu todo o resto. É absolutamente incrível que ninguém havia pensado nisso antes. Eu soube imediatamente que tinha tido uma idéia muito importante. Minha mente estava cheia de possibilidades. "
Como vimos acima, as linhas, colunas e diagonais de um quadrado mágico somam o mesmo número. Se cada número no quadrado é representado por uma linha de tal extensão, segue-se que estes segmentos de linha podem ser unidos de um extremo a outro para formar uma linha maior - e essa linha tem o mesmo comprimento de qualquer linha, coluna ou diagonal que você escolher .
Da mesma forma, se cada número n é substituído por uma forma que n tem espaço, segue-se que as formas em cada linha, coluna ou diagonal, quando colocadas juntas têm a mesma área combinada.
Sallows ainda queria a condição extra de que as formas se encaixam de forma que cada linha, coluna ou diagonal toma exatamente a mesma forma de destino.
"Foi um negócio muito complicado", disse ele. Usando um computador demorei um longo tempo para encontrar formas que trabalhei e quando ele o fiz: "Eu realmente caí da cadeira."
Sallows tinha a impressão de que quadrados Geomagic seriam poucos e distantes entre si. No entanto, ele encontrou milhares deles. "Foi o oposto completo do que eu esperava. Eu estava procurando por algo incrivelmente raro, mas descobri que não era nada raro."
O Eis Shu é o quadrado mágico de 3x3 só que é possível construir quando os números de um a nove são todos usados. (Excluindo rodá-los e refleti-los, que são considerados matematicamente equivalentes.)
Mas quando o Sallows encheu seu quadrado 3x3 com nove Geomagic polnômios que têm áreas de 1-9 - isto é, formas feitas a partir de um quadrado unitário, dois quadrados da unidade, três, e assim sucessivamente até nove quadrados unidade - ele descobriu que existem 1.411quadrados Geomagic , quando a forma-alvo é um retângulo 3x5. Ele foi capaz de encontrar 4.370 quadrados Geomagic onde a forma-alvo é um quadrado de 4x4 faltando uma peça central, 27.110, quando o alvo é um quadrado de 4x4 faltando uma peça de canto e 16.465 quando ao alvo está faltando uma peça não de canto.
Sallows reivindica que suas descobertas lançaram novas luzes sobre os quadrados mágicos. "Ninguém entendeu o que é um quadrado mágico", diz ele. "Nós sempre assumos que são objetos numéricos. A partir dessa perspectiva de que são todos os objectos geométricos, apenas um pequeno subconjunto são objetos numéricos."
Sallows passou a encontrar dezenas de novos tipos de quadrados Geomagic , envolvendo associações de todos os tipos de formas. Ele fez uma galeria de suas descobertas na web no Natal do ano passado, e desde então mais de 34.000 pessoas de 129 países visitaram seu site - tornando-se um hit viral no mundo da matemática recreativa.
Poderia Sallows ver neles alguma utilidade? "Eu não acredito que este material seja da menor importância", responde ele. "É agradável e charmoso e agrada a todos, mas não tem aplicação alguma."
Peter Cameron, da Universidade de Londres, é mais prudente, uma vez que as aplicações são difíceis de prever: ". Euler descobriu quadrados latinos, quando ele estava fazendo uma nova construção de quadrados mágicos, estes agora têm ampla aplicação em muitas áreas, incluindo delineamento experimental. Talvez haja "quadrados geo-Latinos", que tenham alguma aplicação. "
Sejam ou não encontrados usos, contudo, Sallows certamente provou uma coisa: que o fascínio milenar por quadrados mágicos ainda não acabou.
Alex Bellos é o autor das aventuras de Alex em Numberland (Terra dos Números)
Cubos de Rubik e Sudoku
Quadrados Mágicos podem parecer esotéricos, mas seu impacto cultural é evidente quando você abre o jornal ou andar por uma loja de brinquedos. Os dois quebra-cabeças mais populares dos últimos anos - Sudoku e Cubo de Rubik - são conseqüências de uma preocupação de muitos séculos com eles.
No século 18, Leonhard Euler, o maior matemático de seu tempo, foi desenvolvendo maneiras de criar quadrados mágicos. Para fazer isso, ele começou a buscar outro tipo de base que poderia ser usada como uma espécie de modelo para a produção de quadrados mágicos.
O novo conceito de Euler era um quadrado em que cada número ou símbolo, aparece apenas uma vez em cada linha e coluna. Por exemplo:
1 2 3
3 1 2
1 2 3
Embora esses quadrados já fossem conhecidos desde pelo menos alguns séculos antes, Euler foi o primeiro matemático a analisá-los sistematicamente e cunhou o nome "Quadrado Latino".Ele também inventou o conceito-irmão em que em dois quadrados latinos sobrepostos uns aos outros, cada célula da grade é original. Isso ele chamou de "Quadrado Greco-Latino". Por exemplo:
1a 2c 3b
1c 2b 3a
1b 2a 3c
Em 1782, Euler definiu o "problema de 36 agentes", um quebra-cabeça frívolo que levou a um trabalho profundo acadêmico e muitas descobertas. Você pode fazer um Greco-Latino 6x6 compost de seis regimentos de seis agentes cada uma das diferentes fileiras de modo que nenhum se repita em qualquer linha ou coluna? Somente em 1901 foi demonstrado que isso era impossível.
Ao contrário dos quadrados mágicos,Quadrados Americanos e Greco-Latinos encontraram muitos usos e aplicações em ambientes não-matemáticos, por exemplo, desportivos de fixação de listas, criptografia e experimentos biológicos.
A ocorrência mais conhecida de quadrados latinos agora, porém, está em jornais e livros de quebra-cabeça. Sudoku é um quebra-cabeça para completar um quadrado Latino de 9x9 parcialmente concluído que contém os dígitos de 1-9 em cada coluna e linha, com a especificação adicional que os subquadrados 3x3 também contêm os números de um a nove.
A mania do Sudoku anterior foi o Cubo de Rubik, cuja história pode retroceder até o quadrado mágico. Em meados do século 19 em Nova York, Noyes Palmer Chapman, um entusiasta de enigmas amador, fez um modelo físico de um quadrado mágico de tal forma que os números de 1 a 16 ficavam em pequenos quadrados de madeira que poderiam ser encaixados em uma caixa de 4x4.
Ele percebeu que, se ele deixou de fora um dos quadrados, foi possível a deslizar as outras 15 casas ao redor. Isso ficou conhecido como o " Puzzle 15" (Quebra-cabeça 15), que foi moda internacional em 1880 - e é o quebra-cabeça de deslizamento do bloco original, versões que ainda se pode encontrar em lojas de brinquedos.
Em 1970 o designer húngaro Ernö Rubik estava tentando reinventar o Puzzle 15 em três dimensões, quando ele reve a idéia do cubo de Rubik.
A partir do quadrado mágico de Sudoku parece que sempre tivemos gosto por nossa mania de quebra-cabeças de quadrados - embora essa não seja uma questão para os matemáticos, mas para os psicólogos.
Left, a geomagic 3x3 square where each element is a decomino – a polyomino made up of 10 square units. The decominoes in each row, column and diagonal can be fitted together to form a 5x6 rectangle. (They can be assembled in two different ways, which are drawn on either side of each row, column and diagonal.) This is one of only two geomagic squares possible when the pieces are all decominoes - making this type of geomagic square much rarer than when all the pentominoes from one to nine are used. Right, a geomagic 3x3 square where each element is a polyomino, ranging in size from one to nine. The squares around the periphery show how the polyominoes from the corresponding rows, columns and diagonals can be pieced together to form the same shape, in this case a 4x4 square with one internal cell removed. Sallows found 4,370 squares like this, consisting of the polyominoes from one to nine, and making the same target shape - many more than he expected.
A magic square is a grid of numbers for which every line, column and diagonal adds up to the same number. For example:
4 9 2
3 5 7
8 1 6
In this case, the rows, columns and diagonals all equal 15.
The magic square is a simple concept and one that has been around for thousands of years. Mathematicians, artists and mystics have long been fascinated by the mesmerising patterns that they produce, and perhaps it is not surprising that Sallows, under the influence of hallucinogenic drugs, was taken under their spell.
After the acid wore off, however, Sallows's interest in magic squares did not wane. It became an obsession. In fact, he became so fixated on them that after a couple of decades of contemplation he discovered a whole new category of square, which is opening a fascinating new chapter for one of the oldest genres in maths.
Sallows did something very simple, although it had never been done before. He filled his squares with geometrical shapes rather than numbers. Instead of every row, column and diagonal having to add up to the same number, the shapes in each row, column and diagonal had to fit together to create the same master-shape.
He called his invention the "geomagic square". They are an entirely fresh way to appreciate the subtleties of magic squares, and provide beautiful geometric patterns, too.
Perhaps no other area of non-practical mathematics has been so popular for so long as magic squares. Their story begins 4,000 years ago in China, where, according to legend, a turtle crept out of the Yellow River. The reptile is said to have had dots on its underside positioned in such a way as to make the 3x3 square described above.
The Chinese called this square the lo shu, and gave it spiritual importance, believing that it encapsulated the harmonies of the universe. Feng shui, the Chinese system of arranging objects, such as furniture in homes, is in part based on the lo shu.
But veneration of magic squares was not confined to the Chinese. In India amulets with magic squares were worn as protective charms, in Turkey virgins embroidered magic squares on the shirts of warriors and in western Europe Renaissance astrologers equated them with planets.
In 1514, the German artist and mathematician Albrecht Dürer produced his most famous work, Melencolia I, a woodcut portraying a troubled-looking angel surrounded by scientific objects, which included the following 4x4 square:
Some great mathematicians studied magic squares – such as Leonhard Euler in the 18th century, and Édouard Lucas and Arthur Cayley in the 19th – but the field has generally been the domain of passionate laypeople. The most notable aficionado was United States founding father Benjamin Franklin, who liked to spend his spare time constructing particularly innovative variations. In one evening in his 40s he composed a 16x16 square that he claimed was "the most magically magical of any magic square ever made by any magician".
Sallows, aged 66, is very much an enthusiastic amateur. He left school at 17 with no qualifications and his mathematics is entirely self-taught. He was an electronics engineer in the non-academic staff at the University of Nijmegen until he retired two years ago.
While it is certainly remarkable that a non-mathematician has given this established field a new lease of life, it is perhaps only to be expected, since most academic mathematicians would now consider magic squares as too frivolous to occupy their time.
The conventional magic square has n rows and n columns, and must include every number from 1 to n². So, as above, the 3x3 has every number from one to nine and the 4x4 every number from one to 16.
Sallows says he was instantly "turned on by the symmetries" of magic squares, and once he had got the bug he began to invent new rules and modifications. He is celebrated for inventing the "alphamagic square", which is doubly magic:
Sallows gained a certain amount of fame in the recreational maths world for his alphamagic square, but dismisses it as "zany, a bit silly".
He was more excited by looking at the algebra behind magic squares. Édouard Lucas had discovered this formula, which yields a magic square when you substitute a, b and c for numbers:
Sallows was playing around with the Lucas formula: "And then it came to me – why don't I represent the variables by shapes? A 'plus' could be appending, and a 'minus' excising. From that everything else followed. It is absolutely incredible that nobody else had thought of it before. I knew at once I had landed on a really important idea. My mind was crowded with possibilities."
As we saw above, the rows, columns and diagonals of a magic square add up to the same number. If each number in the square is represented by a line of that length, then it follows that these line segments can be joined head to tail to form a larger line – and this line will have the same length whichever row, column or diagonal you choose.
Similarly, if each number n is replaced by a shape that has area n, it follows that the shapes on each row, column and diagonal when put together will have the same combined area.
Yet Sallows wanted the extra condition that the shapes fit together so that each row, column and diagonal made the exact same target shape.
"It was a very complicated business," he said. Using a computer it took a long time to find shapes that worked and when he did: "I really fell of my chair."
Sallows had been under the impression that geomagic squares would be few and far between. Yet he found thousands of them. "It was the complete opposite of what I expected. I was looking for something incredibly rare but discovered it wasn't rare at all."
The lo shu is the only 3x3 magic square that it is possible to construct when the numbers from one to nine are all used. (Excluding squares made from rotating it and reflecting it, which are regarded as mathematically equivalent.)
But when Sallows filled his 3x3 geomagic square with nine polyominoes that have areas from 1 to 9 – that is, shapes made from one unit square, two unit squares, three, and so on until nine unit squares – he found that there are 1,411 geomagic squares when the target shape is a 3x5 rectangle. He was able to find 4,370 geomagic squares where the target shape is a 4x4 square missing one central piece, 27,110 when the target is a 4x4 square missing one corner piece and 16,465 when the target is missing a non-corner edge piece.
Sallows claims his discoveries have cast magic squares in a completely new light. "Nobody ever understood what a magic square is," he says. "We have always assumed they are numerical objects. From this perspective they are all geometrical objects, and only a small subset are numerical objects."
Sallows has gone on to find dozens of new types of geomagic square, involving combinations of all sorts of shapes. He uploaded a gallery of his discoveries on to the web on Christmas Day last year, and since then more than 34,000 people from 129 countries have clicked on his site – making it a viral hit in the world of recreational maths.
Might Sallows's insight have any practical uses? "I don't believe that any of this stuff is of the slightest significance," he replies. "It is delightful and charming and appeals to everyone, but has no application whatsoever."
Peter Cameron, at the University of London, is more circumspect, since applications are hard to predict: "Euler discovered Latin squares when he was giving a new construction for magic squares; these now have wide application in many areas including experimental design. Maybe there are 'geo-Latin squares' that would have some application."
Whether or not uses are found, however, Sallows has certainly proved one thing: that the age-old fascination with magic squares is not over yet.
Alex Bellos is the author of Alex's Adventures in Numberland
The Chinese called this square the lo shu, and gave it spiritual importance, believing that it encapsulated the harmonies of the universe. Feng shui, the Chinese system of arranging objects, such as furniture in homes, is in part based on the lo shu.
But veneration of magic squares was not confined to the Chinese. In India amulets with magic squares were worn as protective charms, in Turkey virgins embroidered magic squares on the shirts of warriors and in western Europe Renaissance astrologers equated them with planets.
In 1514, the German artist and mathematician Albrecht Dürer produced his most famous work, Melencolia I, a woodcut portraying a troubled-looking angel surrounded by scientific objects, which included the following 4x4 square:
16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1
This square is particularly amazing. Not only do the rows, lines and diagonals add up to 34, but the four corners, the four digits in the central square, and the four digits in the top left, top right, bottom left and bottom right quarters do too. There are many other combinations of four numbers in the square that add up to 34, and it is fun looking for them. Dürer even included the year he made the engraving – 1514 – on the bottom line.
Some great mathematicians studied magic squares – such as Leonhard Euler in the 18th century, and Édouard Lucas and Arthur Cayley in the 19th – but the field has generally been the domain of passionate laypeople. The most notable aficionado was United States founding father Benjamin Franklin, who liked to spend his spare time constructing particularly innovative variations. In one evening in his 40s he composed a 16x16 square that he claimed was "the most magically magical of any magic square ever made by any magician".
Sallows, aged 66, is very much an enthusiastic amateur. He left school at 17 with no qualifications and his mathematics is entirely self-taught. He was an electronics engineer in the non-academic staff at the University of Nijmegen until he retired two years ago.
While it is certainly remarkable that a non-mathematician has given this established field a new lease of life, it is perhaps only to be expected, since most academic mathematicians would now consider magic squares as too frivolous to occupy their time.
The conventional magic square has n rows and n columns, and must include every number from 1 to n². So, as above, the 3x3 has every number from one to nine and the 4x4 every number from one to 16.
Sallows says he was instantly "turned on by the symmetries" of magic squares, and once he had got the bug he began to invent new rules and modifications. He is celebrated for inventing the "alphamagic square", which is doubly magic:
five twenty-two eighteen
twenty-eight fifteen two
twelve eight twenty-five
The rows, columns and diagonals add up to 45 when considering the meaning of the words. But when considering the number of letters in each word – so five is 4 and twenty-two is 9 – this also generates a magic square, whose rows, columns and diagonals add up to 21. Another fantastic property of this square is that the word lengths consist of the consecutive numbers from 3 to 11.
Sallows gained a certain amount of fame in the recreational maths world for his alphamagic square, but dismisses it as "zany, a bit silly".
He was more excited by looking at the algebra behind magic squares. Édouard Lucas had discovered this formula, which yields a magic square when you substitute a, b and c for numbers:
a+b a-b-c a+c
a-b+c a a+b-c
a-c a+b+c a-b
When you let a = 5, b = –1 and c = –3, you get the lo shu.
Sallows was playing around with the Lucas formula: "And then it came to me – why don't I represent the variables by shapes? A 'plus' could be appending, and a 'minus' excising. From that everything else followed. It is absolutely incredible that nobody else had thought of it before. I knew at once I had landed on a really important idea. My mind was crowded with possibilities."
As we saw above, the rows, columns and diagonals of a magic square add up to the same number. If each number in the square is represented by a line of that length, then it follows that these line segments can be joined head to tail to form a larger line – and this line will have the same length whichever row, column or diagonal you choose.
Similarly, if each number n is replaced by a shape that has area n, it follows that the shapes on each row, column and diagonal when put together will have the same combined area.
Yet Sallows wanted the extra condition that the shapes fit together so that each row, column and diagonal made the exact same target shape.
"It was a very complicated business," he said. Using a computer it took a long time to find shapes that worked and when he did: "I really fell of my chair."
Sallows had been under the impression that geomagic squares would be few and far between. Yet he found thousands of them. "It was the complete opposite of what I expected. I was looking for something incredibly rare but discovered it wasn't rare at all."
The lo shu is the only 3x3 magic square that it is possible to construct when the numbers from one to nine are all used. (Excluding squares made from rotating it and reflecting it, which are regarded as mathematically equivalent.)
But when Sallows filled his 3x3 geomagic square with nine polyominoes that have areas from 1 to 9 – that is, shapes made from one unit square, two unit squares, three, and so on until nine unit squares – he found that there are 1,411 geomagic squares when the target shape is a 3x5 rectangle. He was able to find 4,370 geomagic squares where the target shape is a 4x4 square missing one central piece, 27,110 when the target is a 4x4 square missing one corner piece and 16,465 when the target is missing a non-corner edge piece.
Sallows claims his discoveries have cast magic squares in a completely new light. "Nobody ever understood what a magic square is," he says. "We have always assumed they are numerical objects. From this perspective they are all geometrical objects, and only a small subset are numerical objects."
Sallows has gone on to find dozens of new types of geomagic square, involving combinations of all sorts of shapes. He uploaded a gallery of his discoveries on to the web on Christmas Day last year, and since then more than 34,000 people from 129 countries have clicked on his site – making it a viral hit in the world of recreational maths.
Might Sallows's insight have any practical uses? "I don't believe that any of this stuff is of the slightest significance," he replies. "It is delightful and charming and appeals to everyone, but has no application whatsoever."
Peter Cameron, at the University of London, is more circumspect, since applications are hard to predict: "Euler discovered Latin squares when he was giving a new construction for magic squares; these now have wide application in many areas including experimental design. Maybe there are 'geo-Latin squares' that would have some application."
Whether or not uses are found, however, Sallows has certainly proved one thing: that the age-old fascination with magic squares is not over yet.
Alex Bellos is the author of Alex's Adventures in Numberland
Hip to be square: Rubik's Cubes and Sudoku
Magic squares may seem esoteric, but their cultural impact is evident whenever you open the newspaper or walk into a toy shop. The two most popular puzzles of recent years – Sudoku and the Rubik's Cube – are both consequences of a centuries-long preoccupation with them.In the 18th century, Leonhard Euler, the greatest mathematician of his day, was devising ways to create magic squares. In order to do this he started looking at another type of square that could be used as a kind of template for producing magic squares.
Euler's new concept was a square in which every number, or symbol, would appear once and only once in each row and column. For example:
1 2 3
3 1 2
2 3 1
While these squares had been known about since at least a few centuries before, Euler was the first mathematician to analyse them systematically and he coined their name "Latin square".
He also invented the sister concept in which two Latin squares are superimposed on each other, and such that each cell in the grid is unique. This he called a "Graeco-Latin Square". For example:
1a 2c 3b
2b 3a 1c
3c 1b 2a
In 1782, Euler set the "36 officers problem", a frivolous puzzle that led to much deep academic work and discoveries. Can you make a 6x6 Graeco-Latin made up of six regiments of six officers each of different ranks so that no rank and regiment is repeated in any row or column? Only in 1901 was it proved that this was impossible.
Unlike magic squares, Latin and Graeco-Latin squares have found many uses and applications in non-mathematical settings, for example in sporting fixture-lists, cryptography and biological experiments.
The best-known occurrence of Latin squares now, however, are in newspapers and puzzle books. Sudoku is a puzzle to complete a partially completed 9x9 Latin square that contains the digits one to nine in each column and row, with the added specification that the 3x3 sub-squares also contain the numbers from one to nine.
The previous puzzle craze to Sudoku was the Rubik's Cube, whose history can also be traced back to the magic square. In the mid-19th century in upstate New York, Noyes Palmer Chapman, an amateur puzzle enthusiast, made a physical model of a magic square such that the numbers from 1 to 16 were on small wooden squares that could be fit in a 4x4 box.
He realised that if he left out one of the squares, it was possible to slide the other 15 squares around. This became known as the "15 Puzzle", which was an international fad in 1880 – and is the original sliding block puzzle, versions of which you can still find in toyshops.
In the 1970s Hungarian designer Ernö Rubik was trying to reinvent the 15 Puzzle in three dimensions when he came up with the idea of the Rubik's Cube.
From the magic square to Sudoku we seem to have always liked our puzzle crazes to come in squares – although this is a matter not for mathematicians, but for psychologists.
A magic square is a grid of numbers for which every line, column and diagonal adds up to the same number. For example:
4 9 2
3 5 7
8 1 6
In this case, the rows, columns and diagonals all equal 15.
The magic square is a simple concept and one that has been around for thousands of years. Mathematicians, artists and mystics have long been fascinated by the mesmerising patterns that they produce, and perhaps it is not surprising that Sallows, under the influence of hallucinogenic drugs, was taken under their spell.
After the acid wore off, however, Sallows's interest in magic squares did not wane. It became an obsession. In fact, he became so fixated on them that after a couple of decades of contemplation he discovered a whole new category of square, which is opening a fascinating new chapter for one of the oldest genres in maths.
Sallows did something very simple, although it had never been done before. He filled his squares with geometrical shapes rather than numbers. Instead of every row, column and diagonal having to add up to the same number, the shapes in each row, column and diagonal had to fit together to create the same master-shape.
He called his invention the "geomagic square". They are an entirely fresh way to appreciate the subtleties of magic squares, and provide beautiful geometric patterns, too.
Perhaps no other area of non-practical mathematics has been so popular for so long as magic squares. Their story begins 4,000 years ago in China, where, according to legend, a turtle crept out of the Yellow River. The reptile is said to have had dots on its underside positioned in such a way as to make the 3x3 square described above.
The Chinese called this square the lo shu, and gave it spiritual importance, believing that it encapsulated the harmonies of the universe. Feng shui, the Chinese system of arranging objects, such as furniture in homes, is in part based on the lo shu.
But veneration of magic squares was not confined to the Chinese. In India amulets with magic squares were worn as protective charms, in Turkey virgins embroidered magic squares on the shirts of warriors and in western Europe Renaissance astrologers equated them with planets.
In 1514, the German artist and mathematician Albrecht Dürer produced his most famous work, Melencolia I, a woodcut portraying a troubled-looking angel surrounded by scientific objects, which included the following 4x4 square:
Some great mathematicians studied magic squares – such as Leonhard Euler in the 18th century, and Édouard Lucas and Arthur Cayley in the 19th – but the field has generally been the domain of passionate laypeople. The most notable aficionado was United States founding father Benjamin Franklin, who liked to spend his spare time constructing particularly innovative variations. In one evening in his 40s he composed a 16x16 square that he claimed was "the most magically magical of any magic square ever made by any magician".
Sallows, aged 66, is very much an enthusiastic amateur. He left school at 17 with no qualifications and his mathematics is entirely self-taught. He was an electronics engineer in the non-academic staff at the University of Nijmegen until he retired two years ago.
While it is certainly remarkable that a non-mathematician has given this established field a new lease of life, it is perhaps only to be expected, since most academic mathematicians would now consider magic squares as too frivolous to occupy their time.
The conventional magic square has n rows and n columns, and must include every number from 1 to n². So, as above, the 3x3 has every number from one to nine and the 4x4 every number from one to 16.
Sallows says he was instantly "turned on by the symmetries" of magic squares, and once he had got the bug he began to invent new rules and modifications. He is celebrated for inventing the "alphamagic square", which is doubly magic:
Sallows gained a certain amount of fame in the recreational maths world for his alphamagic square, but dismisses it as "zany, a bit silly".
He was more excited by looking at the algebra behind magic squares. Édouard Lucas had discovered this formula, which yields a magic square when you substitute a, b and c for numbers:
Sallows was playing around with the Lucas formula: "And then it came to me – why don't I represent the variables by shapes? A 'plus' could be appending, and a 'minus' excising. From that everything else followed. It is absolutely incredible that nobody else had thought of it before. I knew at once I had landed on a really important idea. My mind was crowded with possibilities."
As we saw above, the rows, columns and diagonals of a magic square add up to the same number. If each number in the square is represented by a line of that length, then it follows that these line segments can be joined head to tail to form a larger line – and this line will have the same length whichever row, column or diagonal you choose.
Similarly, if each number n is replaced by a shape that has area n, it follows that the shapes on each row, column and diagonal when put together will have the same combined area.
Yet Sallows wanted the extra condition that the shapes fit together so that each row, column and diagonal made the exact same target shape.
"It was a very complicated business," he said. Using a computer it took a long time to find shapes that worked and when he did: "I really fell of my chair."
Sallows had been under the impression that geomagic squares would be few and far between. Yet he found thousands of them. "It was the complete opposite of what I expected. I was looking for something incredibly rare but discovered it wasn't rare at all."
The lo shu is the only 3x3 magic square that it is possible to construct when the numbers from one to nine are all used. (Excluding squares made from rotating it and reflecting it, which are regarded as mathematically equivalent.)
But when Sallows filled his 3x3 geomagic square with nine polyominoes that have areas from 1 to 9 – that is, shapes made from one unit square, two unit squares, three, and so on until nine unit squares – he found that there are 1,411 geomagic squares when the target shape is a 3x5 rectangle. He was able to find 4,370 geomagic squares where the target shape is a 4x4 square missing one central piece, 27,110 when the target is a 4x4 square missing one corner piece and 16,465 when the target is missing a non-corner edge piece.
Sallows claims his discoveries have cast magic squares in a completely new light. "Nobody ever understood what a magic square is," he says. "We have always assumed they are numerical objects. From this perspective they are all geometrical objects, and only a small subset are numerical objects."
Sallows has gone on to find dozens of new types of geomagic square, involving combinations of all sorts of shapes. He uploaded a gallery of his discoveries on to the web on Christmas Day last year, and since then more than 34,000 people from 129 countries have clicked on his site – making it a viral hit in the world of recreational maths.
Might Sallows's insight have any practical uses? "I don't believe that any of this stuff is of the slightest significance," he replies. "It is delightful and charming and appeals to everyone, but has no application whatsoever."
Peter Cameron, at the University of London, is more circumspect, since applications are hard to predict: "Euler discovered Latin squares when he was giving a new construction for magic squares; these now have wide application in many areas including experimental design. Maybe there are 'geo-Latin squares' that would have some application."
Whether or not uses are found, however, Sallows has certainly proved one thing: that the age-old fascination with magic squares is not over yet.
Alex Bellos is the author of Alex's Adventures in Numberland
The Chinese called this square the lo shu, and gave it spiritual importance, believing that it encapsulated the harmonies of the universe. Feng shui, the Chinese system of arranging objects, such as furniture in homes, is in part based on the lo shu.
But veneration of magic squares was not confined to the Chinese. In India amulets with magic squares were worn as protective charms, in Turkey virgins embroidered magic squares on the shirts of warriors and in western Europe Renaissance astrologers equated them with planets.
In 1514, the German artist and mathematician Albrecht Dürer produced his most famous work, Melencolia I, a woodcut portraying a troubled-looking angel surrounded by scientific objects, which included the following 4x4 square:
16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1
This square is particularly amazing. Not only do the rows, lines and diagonals add up to 34, but the four corners, the four digits in the central square, and the four digits in the top left, top right, bottom left and bottom right quarters do too. There are many other combinations of four numbers in the square that add up to 34, and it is fun looking for them. Dürer even included the year he made the engraving – 1514 – on the bottom line.
Some great mathematicians studied magic squares – such as Leonhard Euler in the 18th century, and Édouard Lucas and Arthur Cayley in the 19th – but the field has generally been the domain of passionate laypeople. The most notable aficionado was United States founding father Benjamin Franklin, who liked to spend his spare time constructing particularly innovative variations. In one evening in his 40s he composed a 16x16 square that he claimed was "the most magically magical of any magic square ever made by any magician".
Sallows, aged 66, is very much an enthusiastic amateur. He left school at 17 with no qualifications and his mathematics is entirely self-taught. He was an electronics engineer in the non-academic staff at the University of Nijmegen until he retired two years ago.
While it is certainly remarkable that a non-mathematician has given this established field a new lease of life, it is perhaps only to be expected, since most academic mathematicians would now consider magic squares as too frivolous to occupy their time.
The conventional magic square has n rows and n columns, and must include every number from 1 to n². So, as above, the 3x3 has every number from one to nine and the 4x4 every number from one to 16.
Sallows says he was instantly "turned on by the symmetries" of magic squares, and once he had got the bug he began to invent new rules and modifications. He is celebrated for inventing the "alphamagic square", which is doubly magic:
five twenty-two eighteen
twenty-eight fifteen two
twelve eight twenty-five
The rows, columns and diagonals add up to 45 when considering the meaning of the words. But when considering the number of letters in each word – so five is 4 and twenty-two is 9 – this also generates a magic square, whose rows, columns and diagonals add up to 21. Another fantastic property of this square is that the word lengths consist of the consecutive numbers from 3 to 11.
Sallows gained a certain amount of fame in the recreational maths world for his alphamagic square, but dismisses it as "zany, a bit silly".
He was more excited by looking at the algebra behind magic squares. Édouard Lucas had discovered this formula, which yields a magic square when you substitute a, b and c for numbers:
a+b a-b-c a+c
a-b+c a a+b-c
a-c a+b+c a-b
When you let a = 5, b = –1 and c = –3, you get the lo shu.
Sallows was playing around with the Lucas formula: "And then it came to me – why don't I represent the variables by shapes? A 'plus' could be appending, and a 'minus' excising. From that everything else followed. It is absolutely incredible that nobody else had thought of it before. I knew at once I had landed on a really important idea. My mind was crowded with possibilities."
As we saw above, the rows, columns and diagonals of a magic square add up to the same number. If each number in the square is represented by a line of that length, then it follows that these line segments can be joined head to tail to form a larger line – and this line will have the same length whichever row, column or diagonal you choose.
Similarly, if each number n is replaced by a shape that has area n, it follows that the shapes on each row, column and diagonal when put together will have the same combined area.
Yet Sallows wanted the extra condition that the shapes fit together so that each row, column and diagonal made the exact same target shape.
"It was a very complicated business," he said. Using a computer it took a long time to find shapes that worked and when he did: "I really fell of my chair."
Sallows had been under the impression that geomagic squares would be few and far between. Yet he found thousands of them. "It was the complete opposite of what I expected. I was looking for something incredibly rare but discovered it wasn't rare at all."
The lo shu is the only 3x3 magic square that it is possible to construct when the numbers from one to nine are all used. (Excluding squares made from rotating it and reflecting it, which are regarded as mathematically equivalent.)
But when Sallows filled his 3x3 geomagic square with nine polyominoes that have areas from 1 to 9 – that is, shapes made from one unit square, two unit squares, three, and so on until nine unit squares – he found that there are 1,411 geomagic squares when the target shape is a 3x5 rectangle. He was able to find 4,370 geomagic squares where the target shape is a 4x4 square missing one central piece, 27,110 when the target is a 4x4 square missing one corner piece and 16,465 when the target is missing a non-corner edge piece.
Sallows claims his discoveries have cast magic squares in a completely new light. "Nobody ever understood what a magic square is," he says. "We have always assumed they are numerical objects. From this perspective they are all geometrical objects, and only a small subset are numerical objects."
Sallows has gone on to find dozens of new types of geomagic square, involving combinations of all sorts of shapes. He uploaded a gallery of his discoveries on to the web on Christmas Day last year, and since then more than 34,000 people from 129 countries have clicked on his site – making it a viral hit in the world of recreational maths.
Might Sallows's insight have any practical uses? "I don't believe that any of this stuff is of the slightest significance," he replies. "It is delightful and charming and appeals to everyone, but has no application whatsoever."
Peter Cameron, at the University of London, is more circumspect, since applications are hard to predict: "Euler discovered Latin squares when he was giving a new construction for magic squares; these now have wide application in many areas including experimental design. Maybe there are 'geo-Latin squares' that would have some application."
Whether or not uses are found, however, Sallows has certainly proved one thing: that the age-old fascination with magic squares is not over yet.
Alex Bellos is the author of Alex's Adventures in Numberland
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