Hooven, Bethel

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Bethel Hooven Union School A MAGIC SQUARE SQUARED College Corner, Ohio-Indiana All magic squares follow a set pattern which may be reflected, translated, or rotated. At the fifth grade level, we learn the pattern for a third order magic square, draw the geometric design, and construct tessellations. The procedure is as follows. LEARN CONSTRUCT '-I 3 8 Cf 5 I , 2 7 ~ Fig. 1. At this point, most fifth-graders have reached their outer-limits, but the lesson may be extended to develop a magic square of huge proportions. Step one - Learn the pattern. Step two - Choose the set of multiples you are going to work with - any consecutive set of multiples will work. Step three - Construct nine consecutive magic squares, each one 3 x 3. You have a choice here. If your first square runs 1-9, your second square may begin with 2 or 10. Choose your sequence and stay with it. If you want to extend this further, I would suggest staying with multiples of one and starting your second square with 10. Step four - Place your nine magic squares on a larger 3 by 3 grid - follow the original pattern. 13 Example 22 2/ i., ]. 7 )..3 ,, lo .2 ~ ).'( 'f 3 8 9 S I 2 7 ~ 13 /2 17 /8 /L/ /0 II /(, l!i Fig. 2. Step five - You now have a 9 by 9 grid which is a magic square made up of nine smaller magic squares. Next, eighty-one consecutive magic squares may be placed within this 9 by 9 grid. Simply look at the square you have just completed and place your new squares in the same sequence. Example Fig. 3. Step six - You have just completed a 27 by 27 magic square containing eighty-one 3 by 3 magic squares and nine 9 by 9 magic squares. NOW, using 729 consecutive 3 by 3 magic squares, place them in the same order as shown on your grid of eighty-one. Good luck! Step seven - I couldn't find a piece of graph paper large enough! Note: This process is infinite. "The Proof is in the Tessellation". Look for the pattern formed by the isosceles triangles and compare their increase in size with the increase in each magic square. 14 USE COMPUTERS WITH CARE Maita Levine and Ray Ro/wing University of Cincinnati Cincinnati, Ohio In a widely used calculus textbook, the student is encouraged to use a computer in the numerical investigation of limits. As an example, students are instructed to calculate (l+x)Vx for x = 0.1, 0.01, 0.001, and so on, until the differences between two successive values of the function differ by less than 0.000001. The text then expresses the hope that the limit will have been found accurate to five decimal places and the following BASIC program is supplied to assist the student: x=l E=2 X = x/10 F = (l+x)i(l/x) PRINT x;F 10 20 30 40 50 60 70 80 90 100 IF ABS (F-E), I.OE - 06 THEN GO TO 90 E=F GO TO 30 PRINT "LIMIT REACHED" END Finally, the student is told that when the program is executed, the following output is generated: x (1 + x) 1/X (rounded to 5 decimal places) 0.1 2.59374 O.Ql 2.70481 0.001 2. 71692 0.0001 2. 71815 10-s 2.71857 10-6 2. 71828 10-7 2.71828 10-s 2. 71828 LIMIT REACHED Unfortunately, on many computers the program doesn't work. The output from an IBM personal computer is given below: 15 

A Magic Square Squared

https://kb.osu.edu/bitstream/1811/86526/1/OJSM_14_Fall1988_13.pdf

A Magic Square Squared

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