From Magic Squares to Sudoku

In 1415, the German renaissance painter and mathematician Albrecht Dürer created this busy copperplate engraving of a woman melancholically contemplating amongst a clutter that includes an hour glass, a balance scale, a sphere, a polyhedron (aka philosopher’s stone), a purse, and keys. Like his contemporary’s Mona Lisa, this mysterious painting has been extensively studied, speculated upon and written about. It is believed that it was planned to be the first in a series of paintings describing various temperaments: melancholic, phlegmatic, choleric, and sanguine. It is titled: “Melencolia I”




Among the peculiar geometrical tool in this painting, in the upper right corner, one can find … a magic square!



Wait; weren’t magic squares invented by Ben Franklin three centuries after?

Magic squares are tables of numbers where sums of every row, every column and both diagonals produce the same number. In Drurer’s square the sum is 34:








Magic squares have been around for thousands of years. Magic squares of size 3x3 are mentioned in the Chinese literature dating 650 BC. Magic squares of size 5x5 and 6x6 are found in an encyclopedia from Baghdad written in the ten’s century. They were engraved in stone and metal, worn as talismans, their usage believed to ensure longevity and prevention of diseases (from Wikipedia). Magic squares can be spotted in art all around the globe. The Indian Parshva temple still contains a 4x4 magic square carved in the 10th century. The Passion facade of Gaudi’s Sagrada Família church in Barcelona, designed by sculptor Josep Subirachs, features a 4×4 magic square with a sum of 33 that is the age of Jesus at the time of Passion.



What has fascinated people about Dürer’s square is that in addition to the regular magic properties, it contained more mathematical patterns and messages:

1) Numbers in each of the following five quadrants add up to 34.



2) Any pair of numbers symmetrically placed about the center of the square (such as these) sums to 17:



3) The two numbers in the middle of the bottom row give the date of the engraving: 1514



4) The numbers 1 and 4 at either side of the date correspond to, in English, the letters 'A' and 'D' which are the initials of the artist.

What about Ben Franklin? As a child and an adult he amused himself with creating large size magic squares such as this 8x8:



While his squares are not truly magic squares because sum of each of the diagonals is not equal to the sum of numbers in every row or column, they are still pretty cool. To compensate for a lack of the diagonal property Franklin defined a broken diagonal or “bended rows” properties where the sum of numbers in each highlighted bended rows is the same.



Magic squares could be considered the ancient relatives of the modern Sudoku game. In the late 19th century, French newspapers started publishing matrix puzzles – 9x9 magic squares with some numbers removed. Then the puzzle slowly evolved taking away the sum requirements and instead requiring that each row, column and diagonal have all the numbers from 1 to 9. And finally, the 3x3 sub-division was added producing the modern day Sudoku (you can click to play):



Thousands of Sudoku are published in the world daily. You have likely seen someone doing Sudoku on his way to work, at the playground, laundromat or in the coffee shop. A good story attesting to Sudoku’s popularity was published in the Sydney Morning Herald. In June 2008 an Australian drug-related jury trial costing over Australian $1,000,000 was aborted when it was discovered that five of the twelve jurors had been playing Sudoku instead of listening to the evidence.

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  12. How I Got My Ex Husband Back...........

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  18. As much as i would like to advice and give my testimony how i resolve my marriage problems, typing a long text wouldn't help me tell the truth better so i would just be straight forward in what i have to say.First love spell are real second if you are to contact any spell caster be very careful so as not to contact the wrong one as at now the only one i would recommend is Ajayi Ololo.He helped me with my marriage problem and some other thing i needed to be fixed can really give details here.At a time it was hard to trust him because he asked for material i had to pay for or produce as wrong as i was,i thought he was reaping me off my money as the non-real and fake spell caster did.Just when i thought to say no more he finished what i had asked him to do for me. I felt humiliated for not trusting him at first.For me i can say he is real and if you ever contact him have no fear you're safe with him because he can fix the longest broken relationship you can see other testimony online if you are skeptical about mine.His contact email is ajayiololo@yahoo.com

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  24. Am so grateful to Dr Sambo for what he has done for me,At first i was thinking he was a scam and also thinking if i was doing the right thing,But when i decided to contact him i told him my problem he laughed and told me that everything will be okay within 48hours,So i decided to give it a try in which i contacted him he told me that i should remove all taught from my mind and be happy,Really on the second day i just got a call from someone.The person was weeping then i decided to ask who was i speaking to then he called his name Richard i was shocked i asked him what happened he said he is really sorry for what he has done to me for the past 3years for leaving me alone.That i should please forgive him that he will make it up with me.So then i said let him come over which he did as he was coming he got me a new car and also said i should have access to his account top prove to me that he will never leave me for anything now we are bought together and we are going to celebrate this Xmas in any country of my choice,Which we are preparing for.Now we are together and he cant do without me friends am so happy to Dr Sambo for bringing back happiness to my life.i swear with my life Dr Sambo is a man to trust you can contact him on his private mail ogunspiritualspelltemple@gmail.com

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  25. My Names is Monique Curry ,AM from United states .i never believed in love spells or magic until i met this spell caster once when i went to Africa in June last year on a business summit i meet a man called Dr ADAGBA, is powerful he could help you cast a spells to bring back my love s gone misbehaving lover looking for some one to love you bring back lost money and magic money spell or spell for a good job i m now happy & a living testimony cos the man i had wanted to marry left me 4weeks before our wedding and my life was upside down cos our relationship has been on for 7 year i really loved him, but his mother was against me and he had no good paying job so when i met this spell caster, i told him what happened and explained the situation of things to him at first i was undecided,skeptical and doubtful, but i just gave it a try and in 6 days when i returned to taxes my boyfriend (is now my husband ) he called me by himself and came to me apologizing that everything had been settled with his mom and family and he got a new job interview so we should get married i didn’t believe it cos the spell caster only asked for my name and my boyfriends name and all i wanted him to do well we are happily married now and we are expecting our little kid and my husband also got a new job and our lives became much better in case anyone needs the spell caster for some help his email address: adagbaspiritualtemple@gmail.com OR adagbaspiritualtemple@yahoo.com you can also call or whatsapp him via +2349057354407 or website;http://adagbaspiritualtem9.wixsite.com/mysite

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    Long term loan (20_40)
    Media term loan(10_20)
    They offer loan like
    Home loan............., Business loan........ Debt loan .......
    Student loan..........,Business start up loan
    Business loan....... , Company loan.............. etc
    Email..........( Abdallah.afandi@financier.com )
    When it comes to financial crisis and loan then Mr Yin Leps loan financial is the place to go please just tell him I Mrs Charina Joseph direct you Good Luck.... HS EMAIL IS : Abdallah.afandi@financier.com ......... Thanks, Contact him asap
    God bless..........

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  27. Why I am writing this testimony is because I made a promise that whosoever that helps me out of my relationship predicament deserves to be known even more by the world. I am 37 years of age, my husband had a very powerful love spell on him that a female he knows put on him and I wanted it removed as this caused serious problems in our marriage and he left me for her. It seemed nothing on earth can bring our paths together forever, I felt really bad because I love him so dearly. I had sleepless nights because I missed my hubby’s touch and care. I seldom visit this webpage but anytime I do, I see two or more positive reviews about a man named dr. Adoda and his mysterious and miraculous works that saves relationships and marriages, this got me curious. Also, I normally see other several reviews, although I learn't that there are lots of scammers out there but I contacted Adoda because I was convinced by a testifier I contacted, that dr. Adoda spell is the big deal. I gave him a chance out of my doubting spirit but to my greatest surprise my husband called me 2 days after dr. Adoda did his thing on him and my hubby’s behalf, that he is so sorry for all the pains he has made me pass through, he wanted to be back home. Presently our love is more than that of Romeo and Juliet, he takes too good care of me now, 110% than before. I know there are others out there, I say if you find dr.adodalovespelltemple@gmail.com then your problem is as well solved. Lydia Ryan W. from California, USA

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  28. Wow i never thought i will happy in my life ever again, i want you all to thank DR OGUDUGU of GREATOGUDUGU@GMAIL.COM, he is a father that takes care of his children the spell caster of our time, he has saved me from a very ugly situation that almost made my life miserable and i want you all to thank him for me,my husband and i have be married for so many years now with children and the relationship was being threatened by a small girl in his working place,she was almost destroying my home i met so many spell casters but to no avail until i met a friend who came from africa and she told me about DR OGUDUGU, and i decided to contact him and try him out as i was losing faith in all spell casters, but he told not to worry that now am in his temple everything will be alright he assured me 100% that his spell has no side effects and that it will work straight and save my home and my marriage and after he has performed the sacrifices and casted the spell, my husband came back to us with love and affection and he loved us more and more and he never looked back or none did his love for me waiver, if you are also experiencing this sort of your situation in your home, if your husband is drifting away, if you are experiencing a broken home or you have lost your husband to a young woman outside there then look no further help is here as GREATOGUDUGU@GMAIL.COM is ready to help you out okay for any of your problem is it financial needs, revenge spell,Hiv cure, job spell, promotion spell what kind of spell do you need DR OGUDUGU is going to solve it for you okay i have promised him to always talk of his good works to the world in general and to any body who cares to listen, you can email him on his personal email on GREATOGUDUGU@GMAIL.COM or his website @ www.greatdrogudugusolutiontemple.webs.com and he will attend to you as soon as possible okay once again his email is GREATOGUDUGU@GMAIL.COM.

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  29. I am Jasmine Jennifer i have used every single spell worker on the internet, spent untold amounts of money and discovered they are all fakes...i was the fool though; doing the same thing over and over again and expecting different results. In the end, I decided that I wanted a tarot reading to know what my future held for me; I contacted a woman who lives locally to me and she told me about a man named (dr odalo); he does not advertise on the internet, has another job for income, has no set prices, makes no false promises and refuses to help anyone that cannot be helped and even helps for free sometimes, he will give you proof before taking money. He is a wonderful man and he was the only person who actually gave me real results. I really hope he doesn't mind me advertising his contact on the internet but I'm sure any help/ extra work will benefit him.whatsapp or call him on+2348063930531 or odalospelltemple@gmail.com He travel sometimes.he is also specialise on ,love marriage,finance, job promotion ,lottery spell, ,Law & Court case Spells,money voodoo,weigh loss voodoo,any sicknesses voodoo,Trouble in marriage,HIV AIDS,it's all he does Hope this helps everyone that is in a desperate situation as I once was; I know how it feels to hold onto something and never have a chance to move on because of the false promises and then to feel trapped in wanting something

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  30. I lost my 1 years relationship three months ago . My ex left me with so many pains and since then I have been heart broken and shattered. I have contact 11 spell casters and 8 of them has rip me off my money without any result. I have Emailed so many sites online looking for a good spell caster till I was directed by a 28 years old girl to DR ADODA who help. At first I never believed him because he was requesting for some amount of money to buy items, it took him three weeks to convince me and something occur to my mind and I said let me give him a try . I was very shocked when Marcus called three days after I sent Dr ADODA HELP the items money. He apologies for all he has done wrong and I am very happy that we are together today because he proposed to me last night that we are getting married next month. I will advise you to contact Email :adodalovespelltemple@gmail.com. because he has done wonders in my life and I believe he can help you out in any problem. FROM MISS Arabella.....

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  31. I feel so blessed again in my marriage after Doctor Oku brought back my husband that separated with me for good 3 mounths. Am Jessica Butler by name from USA. Even though i have mouths all over my body, it won't be enough to thank Doctor Oku for his help upon my life. My husband separated with me for the 3 months and have been in pains and agony without him. So, i searched for help everywhere but nothing worked out not until i meant Doctor Oku who i contacted online. I explained my situation to him and he promised that my husband will get back to me within 24 to 48 hours as far that my heart still beats for him. I believed in him and he prepared a spell for me and my husband called me exactly when Doctor Oku said. He pleaded and said he needs me back and now we are living happily again for the past 9 months. Everyone out there reading my article that needs help should contact him. Email: okutemple@gmail.com or add him on whasapp +2347053113465

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  32. I have my ATM card already programmed to withdraw the maximum of $ 5,000 a day for a maximum of 44 days. I'm so happy with this because I got mine last week and I've used it to get $ 144,000. Mr Martins is giving the card just to help the poor and needy even though it is illegal but it is something nice and it is not like another scam pretending to have the ATM cards blank. And no one gets caught when using the card. I western union some money to him and to my greatest shock, the card was shipped to me in 18hrs.Get yours today ,just send an email to martinshackers22@gmail.com

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