A magic hexagon of order n is an arrangement of numbers in a centered hexagonal pattern with n cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same magic constant M. A normal magic hexagon contains the consecutive integers from 1 to 3n2 − 3n + 1. It turns out that normal magic hexagons exist only for n = 1 (which is trivial, as it is composed of only 1 hexagon) and n = 3. Moreover, the solution of order 3 is essentially unique. Meng also gave a less intricate constructive proof.
A centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. Centered hexagonal numbers have practical applications in materials logistics management.
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. In general where is the side length of the square.
An integer is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 1/2, and √2 are not.
The order-3 magic hexagon has been published many times as a 'new' discovery. An early reference, and possibly the first discoverer, is Ernst von Haselberg (1887).
The numbers in the hexagon are consecutive, and run from 1 to . Hence their sum is a triangular number, namely
A triangular number or triangle number counts objects arranged in an equilateral triangle. The nth triangular number is the number of dots in the triangular arrangement with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, starting at the 0th triangular number, is
There are r = (2n − 1) rows running along any given direction (E-W, NE-SW, or NW-SE). Each of these rows sum up to the same number M. Therefore:
This can be rewritten as
Multiplying throughout by 32 gives
which shows that must be an integer, hence 2n-1 must be a factor of 5, namely 2n-1 = 1 or 2n-1 = 5. The only that meet this condition are and , proving that there are no normal magic hexagons except those of order 1 and 3.
Although there are no normal magical hexagons with order greater than 3, certain abnormal ones do exist. In this case, abnormal means starting the sequence of numbers other than with 1. Arsen Zahray discovered these order 4 and 5 hexagons:
M = 111
M = 244
The order 4 hexagon starts with 3 and ends with 39, its rows summing to 111. The order 5 hexagon starts with 6 and ends with 66 and sums to 244.
An order 5 hexagon starting with 15, ending with 75 and summing to 305 is this:
56 61 70 67 51 55 45 36 48 53 68 74 37 26 29 27 39 73 62 42 33 19 16 31 38 64 58 57 22 20 15 18 23 43 49 63 47 28 21 17 30 34 65 71 35 24 32 25 46 72 59 44 40 41 52 69 54 60 75 66 50
A higher sum than 305 for order 5 hexagons is not possible.
Order 5 hexagons, were the "X" are placeholders for order 3 hexagons, which complete the number sequence. In the upper fits the hexagon with the sum 38 (numbers 1 to 19) and in the lower one of the 26 hexagons with the sum 0 (numbers -9 to 9). (for more informations visit the German Wikipedia article)
39 35 -14 21 -20 -16 -12 37 22 34 -4 X X X -5 -7 -1 36 X X X X -13 -17 30 23 X X X X X -6 24 -21 26 X X X X -3 0 28 -2 X X X 27 -11 -18 25 -15 -9 33 -8 29 31 38 32 -10 20 -19 30 28 -18 -13 -27 -30 -28 18 15 13 12 X X X 27 21 -22 -26 X X X X -11 -24 16 19 X X X X X -12 10 -20 22 X X X X -16 -21 11 26 X X X 20 14 -19 -15 -29 -25 17 24 23 -10 29 25 -17 -14 -23
An order 6 hexagon can be seen below. It was created by Louis Hoelbling, October 11, 2004:
It starts with 21, ends with 111, and its sum is 546.
This magic hexagon of order 7 was discovered using simulated annealing by Arsen Zahray on 22 March 2006:
It starts with 2, ends with 128 and its sum is 635.
An order 8 magic hexagon was generated by Louis K. Hoelbling on February 5, 2006:
It starts with -84 and ends with 84, and its sum is 0.
Hexagons can also be constructed with triangles, as the following diagrams show.
|Order 2||Order 2 with numbers 1–24|
This type of configuration can be called a T-hexagon and it has many more properties than the hexagon of hexagons.
As with the above, the rows of triangles run in three directions and there are 24 triangles in a T-hexagon of order 2. In general, a T-hexagon of order n has triangles. The sum of all these numbers is given by:
If we try to construct a magic T-hexagon of side n, we have to choose n to be even, because there are r = 2n rows so the sum in each row must be
For this to be an integer, n has to be even. To date, magic T-hexagons of order 2, 4, 6 and 8 have been discovered. The first was a magic T-hexagon of order 2, discovered by John Baker on 13 September 2003. Since that time, John has been collaborating with David King, who discovered that there are 59,674,527 non-congruent magic T-hexagons of order 2.
Magic T-hexagons have a number of properties in common with magic squares, but they also have their own special features. The most surprising of these is that the sum of the numbers in the triangles that point upwards is the same as the sum of those in triangles that point downwards (no matter how large the T-hexagon). In the above example,
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, and it is given by the formula
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x + y)n into a sum involving terms of the form a xb yc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example,
In mathematics, the Bernoulli numbersBn are a sequence of rational numbers which occur frequently in number theory. The values of the first 20 Bernoulli numbers are given in the adjacent table. For every even n other than 0, Bn is negative if n is divisible by 4 and positive otherwise. For every odd n other than 1, Bn = 0.
In mathematics, the Fibonacci numbers, commonly denoted Fn form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia (Iran), China, Germany, and Italy.
In geometry, a hexagon is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan, but they are named after Eric Temple Bell, who wrote about them in the 1930s.
In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers.
In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively-defined objects. They are named after the Belgian mathematician Eugène Charles Catalan (1814–1894).
A pandiagonal magic square or panmagic square is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant.
A hexagonal number is a figurate number. The nth hexagonal number hn is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex.
In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base. Square pyramidal numbers also solve the problem of counting the number of squares in an n × n grid.
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement. Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a lattice packing is
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles.
In mathematics, a perfect power is a positive integer that can be resolved into equal factors, and whose root can be exactly extracted. i.e., a positive integer that can be expressed as an integer power of another positive integer. More formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that mk = n. In this case, n may be called a perfect kth power. If k = 2 or k = 3, then n is called a perfect square or perfect cube, respectively. Sometimes 0 and 1 are also considered perfect powers.
In combinatorics, the Eulerian numberA(n, m), is the number of permutations of the numbers 1 to n in which exactly m elements are greater than the previous element. They are the coefficients of the Eulerian polynomials:
The Leibniz harmonic triangle is a triangular arrangement of unit fractions in which the outermost diagonals consist of the reciprocals of the row numbers and each inner cell is the cell diagonally above and to the left minus the cell to the left. To put it algebraically, L(r, 1) = 1/r and L(r, c) = L(r - 1, c - 1) − L(r, c - 1).
In mathematics, a Delannoy number describes the number of paths from the southwest corner of a rectangular grid to the northeast corner, using only single steps north, northeast, or east. The Delannoy numbers are named after French army officer and amateur mathematician Henri Delannoy.
In mathematics, the (2,1)-Pascal triangle is a triangular array.