A Math Forum Web Unit
Mutsumi Suzuki's

Magic Stars: Combining Exchange Rules

Combinations of exchange rules can be confusing, and therefore very interesting.

• If you use Exchange Rule 1 twice, the result will be identical to the original star.

• If you use Exchange Rule 2 and then Exchange Rule 1, the result will be identical to the mirror image of Rule 1.

• If you use Exchange Rule 1 and then Exchange Rule 2, the resulting magic star will be identical to the mirror image of Rule 2.

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A mirror image of a magic star looks like this:

• Simple rotation can not create one of these stars from the other, but we treat them as a single star.

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Followings are examples of transformations, where

1. (o) is the original star;
2. (i) and (i') are mirror images of each other;
3. (ii) and (ii') are also mirror images;
4. (iii) and (iii') are mirror images; and
5. (1),(2), and (3) denote the exchange rule number.

                      (o)
   
                       A
                    B C D E
                     F   G
                    H I J K
                       L
   
               /       |      \
              /(1)     |(2)    \(3)
             /         |        \
   
        (i)           (ii)          (iii)
   
         L             C              D
      C B E D       K A D G        F C A H
       F   G         H   E          B   K
      I H K J       F I L B        E L J G
         A    \        J   \          I
               \            \
         |     |       |     |        |        \
         |(2)  |       |(1)  |        |(1)      \(2)
         |     |       |     |        |          \
               |             |
         B     |       J     |        I            C
      J L E G  |    A K G D  |     C F H A      G D A K
       I   D   |     H   E   |      B   K        E   H
      F H A C  |    I F B L  |     L E G J      B L I F
         K     |       C     |        D            J
   
       (ii')  (3)     (i')  (3)      (i')        (ii')
               |             |                     |
                                                   |(1)
               E             D                     |
            F B L I       H A C F                  
             C   J         K   B                   J
            D A K G       G J L E               D G K A
               H             I                   E   H
                                                L B F I
             (iii')        (iii')                  C
   
                                                  (i)
   

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This map shows the following mirror image relation:

     (i)                                        (i')

      L          L                               J
   C B E D    D E B C    rotate 120 degrees   A K G D
    F   G      G   F   <------------------     H   E
   I H K J    J K H I                         I F B L
      A          A                               C

How many stars?

How many different stars can be derived from one magic star by using or combining the three exchange rules? Three.

One star (the original) plus three stars (derived) = four stars.

Thus the total number of magic stars will be 4 x 20 = 80, where 20 is the number of stars created using the sets described in these Web pages.

Following are just a few examples.