Magic Pentagrams and Hexagrams | Research & Encyclopedia Articles

Magic Pentagrams and Hexagrams

A magic hexagram is a hexagram such that each outer point and each crossing point is assigned a number so that different points are assigned different numbers. The sum of the numbers on any line is required to be the same for all lines. In a pure magic hexagram, the set of assigned numbers is the set of consecutive integers from one to twelve. Finding magic hexagrams is a popular recreation.

In any pure magic hexagram, the sum of the numbers of the corners of each large triangle must be the same. Here is the proof. Let M be the magic number. From the diagram follows these six equations:

• 1. B = M - I - L - J
• 2. C = M - B - A - D
• 3. E = M - C - K - J
• 4. F = M - E - D - G
• 5. H = M - F - L - K
• 6. I = M - H - A - G

Substitute (M - I - L - J) for B is equation 2 to get a new equation for C. Substitute the right side of this new equation for C in equation 3 to get a new equation for E. Continue this until you get a new equation for I. This equation happens to be:

• 7. I = I + 2(L + J + K) - 2(A + G + D).

So, L + J + K = A + G + D.

H.E. Dudeney was the first to attempt to enumerate all pure magic hexagrams. His work appeared in 1926 in the book Modern Puzzles. But, he missed seven hexagrams. Not counting reflections and rotations, there are 80 different pure magic hexagrams. Twelve of these are such that the outer points sum up to the magic number, also. The complement of a pure magic hexagram is formed in this way: if x is the number assigned to a vertex v, then the complement has 13 - x assigned to the same vertex.

In contrast to the hexagram, there are no pure magic pentagrams. That is, the integers from one to ten cannot be assigned to the vertices of a pentagram in such a way that the sum of the numbers on every line is the same.

Here is a proof. If a pure magic pentagram exists, then the sum of all five lines would be equal to five times the magic number. But, each point is in two lines, so this is also equal to two times the sum of all the integers from one to ten. Thus the magic number is 22. The numbers other than 1 on the two lines that contain the number 1 must sum up to 42. But 9 + 8 + 7 + 6 + 5 + 4 = 39, so one of the numbers on those lines must be 10. Similarly, the numbers other than 2 on the two lines that contain the number 2 must sum up to 40. So, one of the numbers on those two lines must be 10. Let L1 be the line containing 1 and 10 and let L2 be the line containing 2 and 10. Then the other numbers on L1 are either {3, 8}, {4,7} or {5,6}. Similarly, the other numbers on L2 are either {3,7} or {4,6}. These two lines cannot intersect more than once, so either (case one) L1 contains 1, 10, 3, and 8 and L2 contains 2, 10, 4, and 6 or (case two) L1 contains 1,10, 5, and 6 and L2 contains 2, 10, 3, and 7. Let L3 be the line containing 3 that does not contain 10. In the first case, L3 must contain a number in L2 that is not 10. But L3 cannot contain any number other than 3 of L1. This is impossible. In the second case, the same considerations lead to one possibility: L3 must contain 6, 4, and 9. Let L4 be the line that contains 4 but not 3. This line must contain 8 since 8 is not in L1, L2, or L3 and every number is in two lines. On the other hand, it must contain either 5 or 1 since it must intersect L1 and it must contain either 2 or 7 since it must intersect L2. Neither of these possibilities will allow the numbers on L4 to sum up to 22, however. So, a pure magic pentagram is impossible.

Weakly magic pentagrams are such that the assigned numbers are either the integers one through twelve with 7 and 11 omitted. There are twelve weakly magic pentagrams not counting rotations and reflections. Almost magic pentagrams are such that only four out of the five lines sum up to the same number. It is also required that the numbers are chosen from the set of integers from one to ten so that each integer appears exactly once. There are seven almost magic pentagrams not counting rotations and reflections.

A prime magic pentagram is such that all the assigned numbers are prime numbers. Here is a smallest such pentagram:

Here is a smallest magic pentagram using consecutive primes:

Magic septagrams, octagrams and so on are also possible. There are 72 pure magic septagrams, 112 pure magic octagrams, and more than 2,000 pure magic nonagrams. These amounts are, of course, only up to rotation and reflection.