A Math Forum Web Unit
Mutsumi Suzuki's

Magic Stars

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Contents || Suzanne's Math Lessons || Magic Squares || Tessellation Tutorials
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Some people read books while commuting, while others listen to music - but Mutsumi Suzuki entertains himself by doing mathematical calculations. Going home on the Shin-Kan-Sen (Japanese bullet train) he began thinking about the magic Star of David. Here are some of his thoughts.

What is a magic star?

A six-sided magic star is constructed using 12 numbers, where all the edges add up to the same number (we will call this the constant N).

Edge A+C+F+H = edge A+D+G+K = edge B+C+D+E = ... = N (constant).

What is the number of the constant N?

Find the sum of all the numbers from 1 through 12:

1+2+3+4+5+6+7+8+9+10+11+12 = 78

therefore A+B+C+D+ ... +L = 78

    A + C + F + H = N
    A + D + G + K = N
    B + C + D + E = N
    B + F + I + L = N
    E + G + J + L = N
 +  H + I + J + K = N
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2 (A + B + C + D+ ... + L) = 6N

therefore N = 2(A+B+C+D+ ... +L)/6 = 2*78/6 = 26

Constructing a magic star using the sum 26

There are many combinations of numbers. Let's see how to begin to list them.

Start with 1 and 2, and subtract (1+2) from the sum 26:

    26 - (1+2) = 23.

Now, since the largest number used will be 12 and we've just used 1 and 2, the remaining edge number must be 11:

    23 - 12 = 11

So the first edge set is {1,2,11,12},
the next set is {1,3,10,12}, and
the next is {1,4,9,12}
. . . .
. . . .

Thus we get

    {1,2,11,12}
    {1,3,10,12}
    {1,4, 9,12}
    {1,4,10,11}
    {1,5,8,12}
    {1,5,9,11}
    {1,6,7,12}
    {1,6,8,11}
    {1,6,9,10}
    {1,7,8,10}

    {2,3,9,12}
    . . .
    . . .
    {5,6,7,8} That's all!

We will use only the first 10 sets.

How do you use these sets?

Let's look at a star that has the number 1 at the top.

On to Using the Sets

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Mutsumi Suzuki
5 October 1996

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Graphics by Sarah Seastone