A Math Forum Web Unit
Mutsumi Suzuki's
Magic Stars
Contents 
Suzanne's Math Lessons 
Magic Squares 
Tessellation Tutorials
Some people read books while commuting, while others listen to music  but Mutsumi Suzuki entertains himself by doing mathematical calculations. Going home on the ShinKanSen (Japanese bullet train) he began thinking about the magic Star of David. Here are some of his thoughts.
What is a magic star?
A sixsided 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
______________________
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:
Now, since the largest number used will be 12 and we've just used 1 and 2, the remaining edge number must be 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
