Magic hexagon

This article describes the mathematical construct magic hexagon. For the economic system of the same name, see Magic Square .

Magic hexagon

A magic hexagon is a hexagonal arrangement of numbers in which the sums of all rows in the three directions give the same value. In particular, analogous to the magic square, the whole numbers, starting from 1, are arranged in the hexagon in such a way that the sums of all rows are the same. Apart from the trivial case in which the hexagon consists of only one number, this is only possible with the side length . ${\displaystyle n=1}$ ${\displaystyle n=3}$

Task

A hexagon with the side length contains numbers and rows in each direction . The identical sum of each row is called the magic number . For the unknown numbers of the hexagon and the magic number, a linear system of equations can be set up. If any whole numbers are allowed as a solution, the system of equations is always solvable, but not unique. ${\displaystyle n}$ ${\displaystyle H=3n^{2}-3n+1}$ ${\displaystyle r=2n-1}$ ${\displaystyle M}$

As a restriction, it is required that the solution numbers are consecutive whole numbers. In particular, a solution with the natural numbers from 1 is sought. Solutions that can be converted into one another by rotating and mirroring the hexagon are counted as one solution.

Solution with the natural numbers from 1

A solution in which the integers from 1 to are arranged in the hexagon only exists for the trivial case and for . In the second case the hexagon has squares and is the sum of the numbers in each row . There is exactly one solution for this, which has been found several times since the end of the 19th century. ${\displaystyle 3n^{2}-3n+1}$ ${\displaystyle n=1}$ ${\displaystyle n=3}$ ${\displaystyle H=19}$ ${\displaystyle M=38}$

In order to deduce for which solutions exist, first the sum of all numbers of the hexagon, i.e. H. the numbers from 1 to , calculated. With you get: ${\displaystyle n}$ ${\displaystyle S}$ ${\displaystyle H}$ ${\displaystyle H=3n^{2}-3n+1={\tfrac {1}{4}}(3r^{2}+1)}$

${\displaystyle S={\frac {1}{2}}H(H+1)={\frac {1}{32}}(9r^{4}+18r^{2}+5).}$

The sum of the numbers in a row is obtained by dividing this total by the number of rows:

${\displaystyle M=S/r={\frac {1}{32}}(9r^{3}+18r+5/r).}$

If this equation is multiplied by 32:

${\displaystyle 32M=9r^{3}+18r+5/r,}$

there is a whole number on the left. So that the right side is also an integer, it must be an integer. This is only possible for only with or . ${\displaystyle {\tfrac {5}{r}}={\tfrac {5}{2n-1}}}$ ${\displaystyle n\geq 1}$ ${\displaystyle n=1}$ ${\displaystyle n=3}$

Solution with consecutive whole numbers

This article or the following section is not adequately provided with supporting documents ( e.g. individual evidence ). Information without sufficient evidence could be removed soon. Please help Wikipedia by researching the information and including good evidence.

If you allow any number of consecutive whole solution numbers, there are generally further solutions. For the sum you have to use the number range from to . For other sums, the following number ranges result with the deviation : ${\displaystyle n\geq 3}$ ${\displaystyle M=0}$ ${\displaystyle -(3n^{2}-3n)/2}$ ${\displaystyle (3n^{2}-3n)/2}$ ${\displaystyle i}$

smallest number:	${\displaystyle i(2n-1)-(3n^{2}-3n)/2=ir-(H-1)/2}$
largest number:	${\displaystyle i(2n-1)+(3n^{2}-3n)/2=ir+(H-1)/2}$
Total:	${\displaystyle i(3n^{2}-3n+1)=iH}$

A formula that gives the largest and smallest for each for which a solution exists is not yet known. ${\displaystyle n}$ ${\displaystyle i}$

In the case there is no solution. ${\displaystyle n=2}$

The solution for shown in the picture above corresponds to the value . There are also solutions for these number ranges: ${\displaystyle n=3}$ ${\displaystyle i=2}$ ${\displaystyle n=3}$

• 001 to 19 with a total of 038: 01 solution
• 0−4 to 14 with a total of 019: 36 solutions
• 0−9 to 09 with the sum of 0: 26 solutions; 14 of these solutions can be converted into one another by completely changing the sign; for the remaining 12, a complete change in sign corresponds to a rotation of 180 degrees. This results in 12 + 7 * 2 (= 26) solutions.00
• −14 to 04 with the sum −19: 36 solutions (all signs changed compared to the solution with a total of 19)
• −19 to −1 with the sum −38: 01 solution (all signs changed compared to the solution with sum 38)

See also

• Magic square
• Mazara

Web links

Commons : Magic Hexagons  - collection of images, videos, and audio files

• math.uni-bielefeld.de