Magic SquaresDate: 10/19/2003 at 15:55:02 From: Thomas Subject: Magic Squares What is the size of the largest magic square that you are aware of that has been constructed without repeating any numbers? I am an amateur magician and a member of American Mensa. I have used the trick wherein the magician guides the audience to the number 34 through some clever use of Hobson's choice or other ruse and then produces a 4 by 4 magic square using the numbers 1 thru 16 which has some twenty modes of symmetry all totaling 34. Having some difficulty in guiding people to the number 34, I devised a method to rapidly calculate and construct a 4 by 4 magic square to total any even number using only integers and to any odd number employing mixed numbers including the fraction one half. This activity may be looked upon as both a monumental waste of time and/or as expanding the breadth of human knowledge. Magic squares can be fun, but are otherwise essentially useless except in magic or for their puzzling nature. I found Ben Franklin's 8 by 8 magic square on your website and got entangled in the logic of its construction. After more analysis than I care to admit, I borrowed some insights from the 4 by 4 square research that I had already done, applied it to the 8 by 8 and in less than five minutes, I had reconstructed his 8 by 8 square using the numbers 1 thru 64 into a new square that also includes a 4 by 4 magic square at its center and has both vertical and horizontal uniformity, but lacks the diagonal uniformity. While pondering this shortcoming, it dawned on me that I could as easily construct a 12 by 12 perfectly symmetrical magic square having at its center a perfect 4 by 4 magic square and with equal ease, I could construct a 16 by 16 perfect magic square containing a perfect 8 by 8 magic square at its center and a 4 by 4 magic square at the center of the 8 by 8 lacking the diagonal symmetry. Logic tells me that this can be expanded to 20 by 20, 24 by 24, 28 by 28 and so on endlessly, limited only by the size of one's paper and the degree to which one is addicted to this lunacy. At this point, the thought occured to me that no matter how big the largest square on record is, I could easily top it, thus attaining some degree of fame for accomplishing the totally useless to a greater degree than some other fool with an idle brain and a fair intellect. Not much seems to have been written on this subject. I strongly suspect that the same logic that I have applied to 4 by 4 squares can similarly be applied to 3 by 3, 5 by 5, 7 by 7 and other prime (or base) numbered squares. Were I to take the time to carefully document my research, it would only tend to demystify magic squares and take the fun out of them, much as the proof that an indeterminate angle cannot be trisected with unmarked straight edge and compass took all of the fun out of that venture (some years after I began to attempt it courtesy of a playful geometry teacher). Two things about Ben Franklin's square completely bewilder me. First is the fact that he took the time to do it at all. I thought him to be a far more practical man, in spite of his notion to fly a kite in a rain storm. Second, the distribution of the numbers follows an almost impenetrable logic and is far different from the simple "cookbook" method that I developed. I doubt that I could reproduce his from memory or by logic, no matter how long I study it. A 4 by 4 square containing the numbers 1 thru 16 is arranged such that 1 is diametrically opposed to 16. 2 is diametrically opposed to 15 and so on. In my 8 by 8, 1 is diametrically opposed to 64. 2 is diametrically opposed to 63 and so on. In a 12 by 12, 1 and 144 are the starting points and for a 16 by 16, 1 and 256 are the starting points. The 16 by 16 totals out to 2,056 in all directions and is very cumbersome to prove. In all cases, these base numbers may be adjusted up or down to produce differing totals or the increment may be changed to alter the upper number and also change the total. Additionally, skipping a range of numbers in the middle allows one to create many different squares to the same total, each containing totally different numbers from the others. Date: 10/19/2003 at 21:38:32 From: Doctor Luis Subject: Re: Magic Squares Hi Thomas, Wow. You've certainly done your research. Unfortunately, nobody can lay claim to having constructed the largest magic square simply because there's a very easy method to construct an N x N magic square where N is an arbitrarily large odd number. I've constructed a 9 x 9 magic square illustrating this method: Get it? You start on the dark orange square in the middle of the top row, place a 1 there, and go up and to the right along the diagonal, putting the next number there. When you can't go up and right, you go down. You keep going until all squares have been filled. And that's it. You can verify that the sums of each row, column, and long diagonal is constant. Can you figure out what the sum is for an N x N square? There's another method when N is a multiple of 4, but I won't go over it right now. Can you think of a method that will work when N is even? Well, I hope you have fun constructing larger and larger magic squares! Let us know if you have any more questions. - Doctor Luis, The Math Forum http://mathforum.org/dr.math/ Date: 10/20/2003 at 14:56:15 From: Thomas Subject: Thank you (Magic Squares) Dear Dr. Luis: I very much appreciate your swift and thorough reply to my question about magic squares. It may surprise you to know that I am also happy to hear that there is no limit to the size of a square prior to my insight. Therefore, there is no need for me to check with the Guiness folks nor to submit anything to them. That makes one less useless thing for me to worry about. Sincerely, Thomas Date: 10/20/2003 at 16:25:23 From: Thomas Subject: Magic Squares Dear Dr. Luis: I erred when I theorized that I could manipulate and expand odd numbered squares in the same manner as even numbered squares. The process is dependent upon the fact that any even numbered square can be divided into two equal number of numbers. No odd numbered square can be so divided as there will always be one number left over, so to speak. It MAY be possible to divide a 3 by 3 square into three groups and then adjust those groups up or down by adding or subtracting some number from each of them, such as taking 1-9 and replacing it with 1- 3, 5-7 and 9-11 or some other such scheme. I don't know and I do actually "have a life" and I really need to get on with it, for the moment, at least. I will likely return to this insanity later. Your example of how to fill in a 9 by 9 square is quite interesting both from the simplicity of the method and from the standpoint of the classiness of the presentation (format of the illustration). I wish that I knew just what software you used to produce the illustration. You mentioned that you were aware of a method of dealing with 4 by 4 squares, but chose not to elaborate upon it. Since I also did not elaborate upon mine, we may both be talking about the same method and not be aware of it or we may have two separate methods. Here is my method. Given the 4 by 4 square: 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 All of the rows and colums add to 34 the diagonals both add to 34 the numbers in the four corners add to 34 the numbers in the four quadrants add to 34 the four numbers in the square in the center add to 34 the two mumbers in the center and the two numbers in the center on the right add to 34 the two numbers in the center on the top and the two numbers in the center at the bottom add to 34 the diagonal pair at the lower left and the diagonal pair at the upper right add to 34 and the diagonal pair at the upper left and the diagonal pair at the lower right add to 34 This amounts to 19 symetrical combinations producing 34. Wow! It may also be noted that each number in the square is diametrically opposed by another number such that their totals will always be 17. So, what? This allows for the independent substitution of another sequential group of numbers to replace the upper half of the numbers. Thus replacing 9 - 16 with 10 - 17 respectively boosts the total for the square to 36 and still retains all of the previously stated symmetries. Neato! Now, if we construct four separate 4 by 4 magic squares using the following sequences: 1 - 8, 57 - 64 9 - 16, 49 - 56 17 - 24, 41 - 48 25 - 32, 33 - 40 We have four separate 4 by 4 squares that each total 130. These four squares may now be joined in any fashion to produce an 8 by 8 square. The resultant square has all of the usual horizontal, vertical, and diagonal symmetry as well as containing totally symmetrical squares in its four quadrants and in addition, a vertically and horizontally symmetrical square at its center. This seems to be the best that I can do, or that can be done. It appears to me that any attempt to satisfy the diagonal symmetry in the 4 by 4 central square automatically destroys the symmetry of the square as a whole. I think that I can prove this, but I am a bit rusty on the logical form of a proof and I think that I will need to use "double reductio ad absurdem" logic to do it by virtue of the seeming evidence that one group of four numbers must be in two places at the same time to do it and since this is absurd, no can do. My conclusions: Odd numbered squares and even numbered squares are totally different animals. I THINK that a 6 by 6 square can be created using some method similar to that which I have used to create the 4 by 4 squares and that it will also be ammenable to expansion. I do not know how many degrees of symmetry I can squeeze out of a 6 by 6 square. At this point, I know how to quickly construct a 4 by 4 square totalling any EVEN number that one might suggest, using only integers, and to any number using fractions - ghastly. Sufficient study of 3 by 3 or 5 by 5 squares will likely yeild a simple method to quickly write squares of those types to any ODD number using only integers. Armed with these two processes, I should be able to rapidly astound an audience by creating a magic square totalling any number they choose almost as fast as I can write the numbers down. Were they still living, my grade school and high school mathematics teachers would be the most astounded. I also sometimes find myself assisting children with mathematics. At times, high school age kids tend to think that they know more than I do and that they need not listen to me. Requesting a number and producing a 4 by 4 square to that total before they have even had the chance to ask why I want it generally gets their attention back. This last seems to be the most beneficial use for this madness that I currently have, unless this can somehow be applied to the process of tying "Lover's Knots" or "Turk's Head" knots. Have I shared anything new to you? Sincerely, Thomas Date: 10/22/2003 at 15:56:06 From: Thomas Subject: Magic Squares Dear Dr. Luis: This topic must be disturbing my sleep. Having studied the 9 by 9 square that you so kindly sent me, I applied its logic to the 3 by 3 types and now can easily recall or recreate the magician's 3 by 3 used in card tricks which I could not do before because of my apparent inability to memorize strings of numbers. Logic sticks, discrete data vanishes. At any rate, I awoke this morning with the notion that I could rebuild the 9 by 9 square from nine 3 by 3 squares. I laid out a 9 by 9 grid and then subdivided it into nine 3 by 3 grids. Starting with the number 1 in the top center small square in the top center large square, I filled in that large square with then numbers 1 thru 9 using the odd square logic for a 3 by 3 square. Then using 3 by 3 logic I moved to the top center small square of the lower right large square and did the same using the numbers 10 thru 18. Then, I moved on to the center left hand large square and so on in simmilar fashion through the rest of the grid finally ending with the number 81 in the center square of the bottom row. The result is a 9 by 9 magic square composed of nine 3 by 3 magic squares with no numbers repeated. I find this square easier to write as it repeatedly employs the 3 by 3 pattern and, therefore, requires less conscious thought. Applying this to magic, I can rapidly take any number, odd or even, subtract 40 from that number and starting with that result, construct a 9 by 9 magic square, having the given number at its exact center. The process is simple, but the effect may be mind boggling on an unsuspecting audience. What do you think? Sincerely, Thomas PS. In case you are curious, I am 60 years old. Date: 10/23/2003 at 17:39:51 From: Doctor Luis Subject: Re: Magic Squares Hello Thomas, Wow. Thank you for your interest. I'm glad to know that you can now create 3x3 magic squares easily. It turns out that a lot of people here ask about magic squares. Dr Jeremiah pointed out to me the wealth of information in our archives, which I'll relay to you. Here are some links of interest, http://mathforum.org/alejandre/magic.square.html as well as some answers involving odd and even squares, Finding Magic Squares http://mathforum.org/library/drmath/view/57967.html Largest Magic Square Ever Known http://mathforum.org/library/drmath/view/56831.html Even-Numbered Even Squares http://mathforum.org/library/drmath/view/56814.html Hopefully these links will serve to answer some of your questions and help you sleep better at night :-) As always, let us know if you have any more questions. - Doctor Luis, The Math Forum http://mathforum.org/dr.math/ |
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