Table of Contents Effect ………………………………………………………………………………………………….. 3 Background …………..…………………………………………………………….…….……….. 3
How the Method Works …………………………………………………………..…...……. 4 The 52 ‘Most Perfect’ Summation Patterns ……………………..…………..….…. 4
Constructing a Panmagic Square ….……………………………….…………….………. 5 General Method (T is EVEN) ……………………………………………………...……...... 7 General Method (T is ODD) ……………….……….……………….…………………….... 9 Summation Patterns When T is ODD ……………………………………….……………. 12 Disguising the Methods ……………………..……………….………………..……..………. 13 Learning Tool .………………………………….……………………… ………………………………….…………………………………………………... …………………………... 13 Final Thoughts .…………………………………….… .…………………………………….……………………………………….……... …………………………………….……... 1 15 5 Contact .…………………………………….………………………………………………….……... .…………………………………….………………………………………………….……... 15 Further Resources Resources .……………………………………………………………………….….…... .……………………………………………………………………….….…... 16 Appendix: Blank Squares ………………………………………………………………………. ………………………………………………………………………. 17
Michael Daniels
Perfectly Possible
PERFECTLY POSSIBLE legant Magic Squares the asy Way
Michael Daniels Effect Performer instantly creates a 4x4 Magic Square for any total freely chosen by the spectator.1 The total can be obtained from the magic square in at least 36 different ways.
Completely impromptu. No set-up or gimmicks.
New, improved method – minimal memory and the simplest of calculations.
Suitable for close-up or stage performances.
Produces elegant magic squares.
Can be immediately repeated for different totals.
Includes a browser application that helps you to learn and practice (Internet connection not required).
Background My ebook Mostly Perfect was was released in 2011. In this, I presented an improved method for the classic demonstration in which a 4x4 magic square is instantly created for any total named by the spectator. While the Mostly Perfect method method is itself quite simple, and produces elegant magic squares, I always felt that an even better method was possible that would require less memory and easier mental calculations. Eventually, I found the solution I had been seeking. Very recently, I discovered that my new method had itself been anticipated (though with more complexity) by the 10th century Indian 2
st
nd
alchemist Nāgārjuna (not to be confused with the 1 /2 century Buddhist philosopher of the same name). I am grateful to Jim Solberg for alerting me to this source. Jim’s recent book on his own original method for constructing 4x4 magic squares is itself a terrific resource and highly recommended for magic square enthusiasts3. I should add that Jim’s construction method is completely different from both the Mostly Perfect method method and from the new procedure to be described here.
1
Total must be at least 34.
2
ṣapuṭ a. See Datta, B. & Singh, A.N. (1992). Magic Squares in India. Indian Journal of History of Nāgārjuna , Kak Science, 27(1) , 51-120. 51-120. http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol27_1_5_BDatta.pdf http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol27_1_5_BDatta.pdf 3 Solberg, James J. (2016). Magic Square Methods and Tricks. Sun Mountain Publications.
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How the Method Works The Perfectly Possible method directly exploits a unique property of the so-called ‘most perfect’, ‘diabolic’, ‘panmagic’ or ‘super-magic’ squares. Beyond meeting the basic requirement of a magic
square – i.e., that all the rows, columns and both diagonals add up to the same total (the magic constant ) – ‘most perfect’ squares achieve this total in 52 different symmetrical ways, as shown in
the diagram below.
The 52 ‘Most Perfect’ Summation Patterns
The explanation for this extraordinary degree of flexibility is the pandiag pandiagonal onal structure structure of these squares. This simply means that every pair of cells that are diagonally two apart add up to the (which is exactly half the square’s magic constant). same number (which
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Consider, for example, the following well-known 4x4 ‘most perfect’ magic square used by Harry Lorayne as the basis for his celebrated ‘Instant Magic Square’ presentation.4
8 11 14 1 13 2 7 12 3 16 9 6 10 5
4 15
The magic constant (T) for this square is 34, and it can c an be achieved using the 52 patterns indicated above. Now note how EVERY two cells that are two apart on ANY diagonal add up to T/2 = 17, as shown in these three examples.
That’s it! That’s the entire secret of the most-perfect squares.
Constructing a Panmagic Square The challenge for those who want to generate these squares is how to ensure that a pandiagonal structure is achieved. Before reading any further, you might like to try to create such a square yourself – for example, where the magic constant (T) is 40. If you don’t have a proper method, you will find it very difficult indeed. The method I will describe is very easy and can be learned in a few minutes. All you need to do is to remember a simple sequence of eight cells and then make the simplest of mental calculations. This involves nothing more complicated than dividing by two, then adding (or subtracting) two numbers at a time.
4
Harry Lorayne (1977). The Magic Book . New York: G.P. Putnam’s Sons & London: W.H. Allen.
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First, you need to commit to memory the following sequence of eight cells. The pattern of this sequence will always remain the same, no matter what total you are seeking to achieve.
Mentally walk through this sequence from 1 to 2 to 3 etc. until you have it firmly fixed in your memory. It might help to note that you move down the rows, then up, each time making a knight’s chess move followed by a diagonal move (except for the bottom row where you shift one cell left from 4 to 5). The fact that the sequence creates a symmetrical pattern ( outer-inner-outer-inner ) also makes it very easy to learn. You should now be able to complete easily the rest of Harry Lorayne’s magic square (where T=34) by filling in the blank cells with the pandiagonal complements of T/2 (=17). In other words, you should enter the complement of each cell in the cell that is two apart diagonally, as shown below.
The pandiagonal principle of entering complementary values in cells that are two apart diagonally is the key to understanding the Perfectly Possible method. Note that although these procedures will work no matter how large the total, to keep your mental calculations as manageable as possible I recommend that you only work with two digit totals (i.e., always ask the spectator to choose a total less than 100). Not only will this simplify your task, but it will also be much easier for spectators to follow the summation patterns when you demonstrate these at the end.
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General Method (T is EVEN) When T is any EVEN number (it should be at least 34), constructing the square for that total can be done very simply. The most straightforward approach is to keep the first eight digits the same as before and simply replace the numbers in the eight blank cells with the pandiagonal complements of T/2. For example, if T=76, first fill in the numbers 1-8 using the standard sequence. Then calculate T/2 = 38. Finally, fill in the eight blank cells with the pandiagonal complements of 38. This would produce the following square.
Although this method works correctly, the resulting square is less than elegant because the eight largest values differ noticeably in magnitude from the eight smallest values. Not only is this aesthetically jarring, but it may also give the astute spectator clues to how the square has been generated. To avoid this problem, and to produce more balanced and elegant squares, we need to change the values in the first eight cells. However, while their values must change, their sequence does not. We simply start the sequence at a new number. To determine the best starting value (for the top right cell), first calculate T/2. Next, mentally split T/2 into two parts (A and B) where A minus B is at least 15 . You can split anyway you like, although the resulting square will look most balanced if you aim for the difference between A and B to be close to (but not less than) 15. The reason that A-B must be at least 15 is to avoid duplication of numbers in the final square. Having calculated a good value for A and B, enter the lower value (B) into the starting cell . Then increment each of the next seven cells in the memorised sequence (increasing the values by 1 in each successive cell). Finally, enter the pandiagonal complements of T/2 into the eight remaining cells. You are then done! For example, if T=76, then we can split T/2 (=38) in various ways, such as 30:8, 29:9, 28:10, or 27:11. However, we cannot split 26:12 or 25:13 etc. because the difference between the two values is less than 15.
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It does not matter which of the acceptable splits we choose to work with – each will produce a most perfect square with the correct magic constant. Let us assume we choose 28:10. To complete the square, we first write 10 in the starting cell, then increment by one each of the successive seven cells in the memorised sequence. Finally, we simply enter the pandiagonal complements of T/2 in the eight remaining blank cells, as seen here.
I think you will agree that this produces a much more balanced and pleasing square than the previous one. Note that we could just as successfully have chosen other splits for T/2. Here are some examples.
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General Method (T is ODD)5 When the magic constant (T) is an ODD number (it should be at least 35), constructing the square is a little more complicated because T/2 is a fractional number. For example, if T=65, then T/2 will be 32.5. While we could complete complete the square in the way already described, using fractions in some cells, this would produce a very ugly result. Fortunately, there is a straightforward solution to this problem. All we need to do is to alternate rounded-up and rounded-down values in certain cells. This is how it works: First, we add 1 to the required total, giving an EVEN working number. For example, if T=65, we will work with 66. Let’s call this value N.
Next, we calculate N/2 = 33. We then split this value into two parts (A and B) so that A minus B is at least 166. As before, you can split anyway you like, although the resulting square will look most balanced if the difference between A and B is close to (but not less than) 16. This will ensure that you do not get duplicate numbers in the final square. Having calculated a good value for A and B, B , enter the lower value (B) into the starting cell , as previously described. Then increment each of the next seven cells in the memorised sequence (increasing the values by 1 in each successive cell). For example, if T=65 (N=66), then we can split N/2 (=33) in various ways, such as 28:5, 27:6, 26:7, or 25:8. However, we cannot split 24:9 or 23:10 23:1 0 etc. because the difference between A and B is less than 16. Let us assume we choose 27:6. To complete the square, we first write 6 in the starting cell, then increment by one each of the successive seven cells in the memorised sequence, exactly as we did for EVEN total squares. This will give us the following semi-completed square.
5
If you do not want to learn the method for constructing odd-total cells, you could always FORCE an even
total. The simplest way to do this is to ask the spectator to think of a number (say, between 20 and 50) and then to double it (‘to make it even more difficult’). 6 Note that the difference between A and B must be at least 15 for EVEN totals and 16 for ODD totals.
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At this point, I draw your attention to the pattern formed by the remaining blank squares. I have coloured these red here.
This is another easy pattern to remember because it is again symmetrical – in fact, it is simply the green pattern flipped on its head. It also resembles a pixelated human figure having a head, arms, hips and feet (let’s call him ‘blockman’).7 Like the original green starting pattern, the blockman pattern also represents a sequence of cells, but this time we start our sequence on the bottom row.
Note that this sequence follows the same pattern as our original sequence, but now we move up the rows, then down, each time making a knight’s chess move followed by a diagonal move (except for the top row where we again shift one cell left from 4 to 5). Also, as before, the pattern is outerinner-outer-inner.
It is crucial that you learn and remember this sequence, because it makes entering the correct values for the remaining eight cells in odd-total squares very easy indeed. We just apply two straightforward rules:
7
I am grateful to Jim Solberg for pointing out this useful aide-memoire.
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Rule 1: Enter the complements of N/2 in the first four cells of the blockman sequence (i.e., moving up the rows). Rule 2: Enter the complements of N/2 N/2 – 1 in the final four cells of the blockman sequence (i.e., moving down the rows. In other words, using these rules, you should enter the complement of each of the eight alreadyentered cells in the cell that is two apart diagonally. To help you follow f ollow along, I will colour the four cells that complement N/2 in yellow and the four cells that complement N/2 – 1 in blue. If you follow these rules correctly, you will end up with this magic square.
Before reading any further, you might like to remind yourself of the procedure and then practice constructing a few odd-total squares. Why not try the following before you look at the solutions that are shown below? 1.
T = 51, N = 52, N/2 = 26, A:B = 22:4
2.
T = 73, N = 74, N/2 = 37, A:B = 27:10
3.
T = 85, N = 86, N/2 = 43, A:B = 30:13
Solutions
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Summation Patterns when T is ODD Because the magic squares constructed when T is odd are not fully pandiagonal8, not all 52 ‘most perfect’ summation patterns shown previously will work. However, if you construct the odd total
squares using the above method, you are certain to obtain the 36 patterns shown in green below.
Green patterns apply for all values of T (ODD and EVEN) Orange patterns apply only when T is EVEN
8
The squares are not truly pandiagonal because not every pair pair of cells that are two apart diagonally sums to the same value. Half of the pairs sum to N/2, while the other half sum to N/2-1 .
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Disguising the Methods I recommend that, when first learning the Perfectly Possible methods, you follow the above procedures exactly – entering each cell’s number in the t he order I have described. Once you are confident that you understand the procedures for both even and odd totals, and can complete any square without error, you should practice mixing things up. This is important in actual performance, to disguise the constructional methods. The easiest way to disguise what you are doing is to write in the first eight numbers in a random r andom order. Simply make a mental note of the sequence needed, but mix up the order in which you enter the values in each cell. For even better disguise, try entering some of o f the complementary values at the same time as you are writing in the initial sequence of eight numbers. This is very easy to do when T is EVEN. You must take more care when T is ODD. When disguising your method, the most important thing is to aim for accuracy when completing the square. It can be quite difficult (and embarrassing) to correct errors, so take things carefully. car efully. Speed, on the other hand, is much less important. In fact, because you need to calculate values individually for many of the cells, this will itself place a limit on how fast you can go. In my opinion, it can be a presentational advantage to be forced to calculate individual values. Spectators will notice that you are thinking carefully as you are completing the square, and this makes the whole process much more believable as a demonstration of mental skill and agility. Compare this, for example, with Harry Lorayne’s method, which can almost be done on autopilot and, if not acted convincingly, could appear suspiciously too easy.9
Learning Tool To help you learn and practice constructing the squares, this ebook comes with a training app that runs in your web browser. Once you have downloaded the file (pptrainer.html) you no longer need to be connected to the Internet to use it – simply open the file from your local device. You should then see this screen:
9
Harry Lorayne’s method also has the disadvantage di sadvantage that it can result in extremely imbalanced squares. Furthermore, it cannot be repeated with a different total for the same audience.
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The learning tool allows you to choose either EVEN or ODD totals to practice with. Click the button of choice. You will then be shown a random total for the square (possible values are 34 to 99). Mentally calculate and enter the numbers for ALL cells, then click the CHECK button.
If your square is correct10, you will be congratulated. Otherwise you will get an alert explaining your error, followed by an ‘OOPS!’ message.
Can you spot what caused the duplication of numbers in the incorrect square?
(Answer given in the footnote) 11
10
The square must achieve all 52 summation patterns for EVEN totals, or all 36 summation patterns for ODD
totals. 11 The incorrect square utilizes a 20:6 split between A and B. The difference between these values is 14, which is less than the minimum of 16 that is needed for odd total squares.
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Final Thoughts That’s all you need to know about the Perfectly Possible methods for constructing instant magic
squares for any total. There are more advanced ways of applying and extending the principles I have described, but I have deliberately omitted these to keep everything as simple and straightforward as possible. In my opinion, apart from the Harry Lorayne method, Perfectly Possible is the easiest and most practical approach to the instant magic square. I hope that you find f ind it useful. When presenting the effect, you should normally explain in advance what a magic square is, how they have been considered to have magical properties throughout the ages, how all the numbers should be different, and how they are very difficult to construct, especially when the total can be any value at all12. Make construction of a magic square sound almost impossible, especially when done instantly! One great advantage of the instant magic square is that it can be performed completely impromptu, provided you have a pen and paper. However, if you want something a bit ‘showier’, you can carry some pre-printed blank squares around in your wallet. You will find templates that you can print out in the Appendix, including some larger size squares which are useful when performing for groups. If you want to repeat the effect, you could laminate l aminate these templates and write with an erasable marker. For stage performance, a markerboard or flip chart is best.
Contact If you have any comments or questions about Perfectly Possible, or any of my other effects, or wish to share your ideas on any aspects of methodology or presentation, you can email me at
[email protected] [email protected] I look forward to hearing from you.
Mike Daniels www.mindmagician.org www.lybrary.com/michael-daniels-m-63273.html
First Edition Copyr igh t © Michael Daniels, December 2016. 2016.
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For some interesting facts and ideas, you might like to watch my 2014 TEDx talk on The Magic, Myth and Math of Magic Squares https://youtu.be/-Tbd3dzlRnY https://youtu.be/-Tbd3dzlRnY
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Further Resources Books Andrews, W.S. (1917). Magic Squares and Cubes , 2nd edition. Open Court Publishing Company. [Republished by Dover, 1960]. Corinda, T. (1984). 13 Steps to Mentalism. Supreme Magic Company. (pp. 65-66). www.lybrary.com Dalal, S. (2011). Patterns of Perfection Revisited . www.lybrary.com Daniels, M. (2011). Mostly Perfect . www.lybrary.com www.lybrary.com Dexter, W. (1974). Feature Magic for Mentalists. Supreme Magic Company. (pp. 95- 102). www.deceptionary.com Dyment. D. (2008). Stimulacra. 2nd (revised) printing. www.deceptionary.com Dyment. D. (2011). Mindsights. 2nd (revised) printing. www.deceptionary.com www.deceptionary.com www.deceptionary.com Dyment, D. (2013). Idiopraxis. www.deceptionary.com Farrar, M.S. (2006). Magic Squares. http://www.MagicSquaresBook.com/ http://www.MagicSquaresBook.com/ Fulves, K. (1983). Self-Working Number Magic. Dover. Gardner, M. (1966) More Mathematical Puzzles and Diversions. Pelican Books. (Chap. 12). Heath, R.V. (1953) Mathemagic. Dover. Lorayne, H. (1977). The Magic Book . New York: G.P. Putnam’s Sons & London: W.H. Allen. (pp. 218222). Lorayne, H. (2006). Mathematical Wizardry . www.harryloraynemagic.com www.harryloraynemagic.com Meyer, O. (1961). The Amazing Magic Square and Master Memory Demonstration. www.lybrary.com www.lybrary.com Miller, W. (2009-2013) E-Z Square 1-6. www.lybrary.com www.lybrary.com Ollerenshaw, K. & Bree, D. (1998). Most Perfect Pandiagonal Magic Squares: Their Construction and Enumeration. The Institute of Mathematics and its Applications. Solberg, J.J. (2016). Magic Square Methods and Tricks. Sun Mountain Publications.13 www.lybrary.com Wasshuber, C. (nd). The Ultimate Magic Square (That's Magic) . www.lybrary.com
Websites Collection of Magic Squares and Figures. Figures. www.taliscope.com/Collection_en.html www.taliscope.com/Collection_en.html Magic Square (Wikipedia). (Wikipedia). http://en.wikipedia.org/wiki/Magic_square http://en.wikipedia.org/wiki/Magic_square Magic Squares (Mark Farrar). Farrar). www.markfarrar.co.uk/msfmsq01.htm www.markfarrar.co.uk/msfmsq01.htm Magic Squares Squares http://www.magic-squares.net/ http://www.magic-squares.net/
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Includes a comprehensive list of books, articles, online material, DVDs and tricks.
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Appendix
Blank Squares (Print, cut and laminate as required)
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