The Magic Cross and Other Magic Figures

The Magic Cross

My latest contribution to magic figures is one of the simplest possible ones. It is a square of 3 rows and 3 columns, but without a center cell. I conceived this when I saw an inspiring pattern in a floor tile arrangement. I called it the Magic Cross.

Magic figures are called ‘normal’ when they are filled with consecutive integer numbers starting from 1.

The only solution for the normal Magic Cross, apart from its reflections and rotations, is easily found, since the arms of the cross need to add up to 12 (the magic constant). We can achieve this only by adding 5+7, and 8 + 4.

The Magic Cross unique solution is the following:

The Magic Cross has only eight cells, filled with the numbers 1 to 8, and clearly exhibits seven sums adding up to its magic constant: Each of its 3 rows and its 3 columns add up to 12, and the sum of both of its diagonals, which are the vertices of the square (1,2,3 and 6), also adds up to 12.

However, looking at the Magic Cross more carefully, we can find another four left-right symmetric patterns that add up to 12:

• The number on the left arm of the cross (5), plus the number on the bottom arm (4), plus the number on their opposite vertex (3), totals 12.
• The number on the right arm of the cross (7), plus the number on the bottom arm(4), plus the number on their opposite vertex (1), totals 12.
• The number on the left arm of the cross (5), plus the number on the top arm (8), minus the number on the vertex between them (1), totals 12.
• The number on the right arm of the cross (7), plus the number on the top arm (8), minus the number on the vertex in between them (3), totals 12.

We can also observe that, diagonally:

• The numbers in the opposite vertices of the square are one the double of the other: (1,2) (3,6).

As a bonus:

• The numbers in the vertical arms of the cross are also one double of the other (4,8).

From left-to right instead:

• The top and bottom corners are the triple of each other: (1,3), (2,6).

Amazingly, we can see one more property that is valid for all the arms of the cross:

• The numbers on each arm of the cross (8,5, 7 and 4) are the sum of the two opposite vertices of the square! That is, 8 is 6 + 2; 5 is 3 + 2; 7 is 6 + 1, and 4 is 1 + 3.

Other Simple Magic Figures

Magic squares

The most famous, and the smallest possible normal magic square (excluding the trivial single cell square), is the 9-cell square, with three columns and three rows.

It is easy to demonstrate that the 4-cell square of two rows and two columns has no solution.

In all magic squares the sum of each column, row and diagonal must add up to the same magic constant. In our 9-cell normal magic square this is 15.

So, the normal 9-cell magic square is filled with nine numbers and exhibits eight sums adding up to its magic constant:

The magic constant can be easily calculated by adding up the all of the numbers in the square, which is the series 1+2+…9 = (9 x 10)/2 = 45, and dividing this by the number of rows: 3.

Thus, the general formula, valid for any size magic square with n rows and columns, is n2(n2+1)/2n, or: n(n2+1)/2.

If we add the same number to all its cells, it remains a magic square, but it is not considered ‘normal’. The magic constant, will be the original plus n times the number we added to each cell.

The classic 9-cell magic square, filled with numbers from 1 to 9, has only one solution, apart from its reflections and rotations.

If we consider squares bigger than n = 3, there are more than one solution for each square.

There are methods for creating magic squares of any size bigger than n = 3. There are various sources in the literature for such methods.

However, a unified single method, for creating all squares of any size bigger than n=3, or n= 4, has not been found.

The Simplest Magic Triangle

Terell Trotter, in the early 1970s proved that there are only four solutions for the simplest (smallest order) normal magic square. This figure is filled with six integer numbers, the numbers 1 to 6, and exhibits three sums adding up to a magic constant of 9, 10, 11 or 12, depending on how the same numbers are arranged:

The Simplest Magic Hexagon

The following solution is the simplest (smallest order) for magic hexagons, and is the only solution for this hexagon with 19 cells, filled with the numbers 1 to 19. In this Magic Hexagon all Sums in all three directions add up to 38:

The Magic Pulsar

When I discovered the following magic figure, I named it “The Magic Pulsar” because its magic constant in all three directions, is either 32 or 33 depending on how the numbers, 1 to 12, are arranged, and because it has only two unique solutions:

Only later I realized that an equivalent of this magic star had been discovered earlier by two scientists from Israel.

The Simplest Magic Star

This magic figure has six vertices and six intersections. A number is assigned to each intersection.

This was found by H. E. Dudeney in 1926. The normal magic star with six vertices has 80 solutions. Like the previous Magic Pulsar, it is also filled with the numbers 1 to 12. The sums of the numbers on each line, in three directions, add up to its magic constant 26.

There are many scientific papers on the mathematics of magic stars.

The fruit of Life

The “Fruit of Life” is the name given to a geometrical figure of 13 circles assembled along the three main directions at 60 degrees angles, thus creating a composite structure analogous to a hexagonal shape.

This Fruit of Life figure is a subset of another hexagonal figure (called by the same sources the “Flower of Life”) composed by 19 non-intersecting circles, disposed in a hexagonal formation.

I discovered the fruit of life figure a few years ago. I then filed it with consecutive numbers from 1 to 13, and found its unique, magic normal solution below, with a magic constant of 35:

This figure has many amazing properties, which I describe in a separate article at: https://bit.ly/3aPmQmZ

The Magic Rhombicube

I was inspired to work on this figure when I looked at a tile arrangement of a floor in Pompei.

The normal, Magic Rhombicube is filled with the twelve integer numbers 1 to 12, and displays twelve sums adding up to its magic constant: 26.

A solution for the smallest, non trivial rhombicube, one of the second order, is the following:

Again, the Magic Rhombicube has many amazing properties, as described in my article at: https://bit.ly/2wAJpg4

If you had to vote, which of the Magic Figures described in this article do you like best?