Transform Any Addition Magic Pentagram Into A Multiplication Magic Pentagram

Pentagram is a pentagon where the sides are extended until they intersect to form the shape of a five-pointed star. There are ten intersections and five line segments in a pentagram. Each line segment consists of four vertices or intersections and two lines at each point.  When you put numbers at each intersection and the sums of the five vertices are equal, then the pentagram is an addition magic pentagram. Figure 1 shows an example of an addition magic pentagram with a magic constant,  S, which is equal to 24 in this case, using the numbers 1 through 12 except for 7 and 11 (which is easy to remember, right?). This is the smallest possible addition magic pentagram and there is no solution for using consecutive numbers.

addition magic pentagram.jpg
Fig. 1: S = 24

But how can you construct a multiplication magic pentagram based on the above addition magic pentagram? The following method can be applied to any addition magic pentagrams to transform them into multiplication magic pentagrams. Here are the steps:

  1. You need to designate a constant x, which can be any number higher than 1 and rewrite all the numbers in the magic pentagram into exponents of your constant. Thus, starting with 1, replace all the numbers in a clockwise order by x1, x12, x4, x9, x2, x6, x5, x8, x3, and x10, as in Figure 2.
Fig. 2
  1. Assign a constant to x, any numbers greater than 1 can be used. For simplicity, we’ll use 2.
  2. Calculate the new numbers. The magic constant of the multiplication magic pentagram is the constant to the power of the magic constant of the addition magic pentagram S or xS. In our example x = 2 and = 24, hence, x= 224 which is equal to 16,777,216.
Fig. 3: Multiplication Magic Pentagram with a magic constant of 16,777,216

All of the segments of the magic pentagram in Figure 3 have the same products (16,777,216). As stated earlier, this technique can be applied to other magic pentagrams as well such as to these addition magic pentagrams in Figure 4 and Figure 5 which have magic constants of 32 and 40 respectively.

addition magic pentagram s=32.jpg
Fig. 4
Fig. 5

You can try to convert these addition magic pentagrams into multiplication magic pentagrams, but be warned that the numbers would be quite large so you are advised to use a computer software (like Wolfram|Alpha) to calculate them.


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My name Edmark M. Law. I work as a freelance writer, mainly writing about science and mathematics. I am an ardent hobbyist. I like to read, solve puzzles, play chess, make origami and play basketball. In addition, I dabble in magic, particularly card magic and other sleight-of-hand type magic. I live in Hong Kong. You can find me on Twitter` and Facebook. My email is

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