If you come from mathematics, as I do, you realize that there are many problems, even classical problems, which cannot be solved by computation alone.
Roger Penrose
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Magic circles have fascinated mathematicians, magicians, and mystics for centuries due to their captivating blend of mathematical symmetry and mystical allure. A magic circle is a geometric arrangement of natural numbers placed in a circular pattern, where the sums of the numbers along specific paths – such as radii, concentric rings, or arcs – are equal. This concept mirrors the principles of magic squares but comes with the added complexity and beauty of radial symmetry.
Historically, magic circles have been connected to numerology, ancient art, and even astrology. However, their mathematical significance lies in their unique properties and the challenge they present in construction and analysis. They are a testament to the interplay between creativity and logic, offering endless possibilities for exploration.
In this blog post, we will consider the following two examples of magic circles.
Example – I
Consider the following diagram, which contains 33 small vacant circles. The task is to fill these circles with the natural numbers from 1 to 33 (without repetition) such that the sum of the numbers along every line is the same.
Magic circles were invented by the Chinese mathematician Yang Hui. One of his magic circles is constructed using the natural numbers from 1 to 33, arranged on four concentric circles, with the number 9 placed at the centre. This particular arrangement represents a magic circle of order 4. A solution for this magic circle is provided below.
The following facts are apparent from the figure:
1. The sum of the numbers on the four diameters is 147. There are four diameters, and the four sums are, respectively, as follows.
28 + 5 + 11 + 25 + 9 + 7 + 19 + 31 + 12 = 147
27 + 15 + 3 + 24 + 9 + 30 + 14 + 21 + 4 = 147
20 + 16 + 23 + 10 + 9 + 2 + 29 + 32 + 6 = 147
33 + 1 + 13 + 22 + 9 + 18 + 26 + 17 + 8 = 147
2. The circle contains eight radii (as shown in the figure), and the sum of the numbers in these eight radii, excluding 9, equals the magic number 69.
28 + 5 + 11 + 25 = 69
7 + 19 + 31 + 12 = 69
27 + 15 + 3 + 24 = 69
30 + 14 + 21 + 4 = 69
20 + 16 + 23 + 10 = 69
2 + 29 + 32 + 6 = 69
33 + 1 + 13 + 22 = 69
18 + 26 + 17 + 8 = 69
Moreover, the sum of all the numbers along the diameters, excluding the number 9, is 2 × 69 = 138.
3. The sum of each of the eight numbers located on the circumference of the four circles, when added to 9, located at the centre, equals 147.
9 + 28 + 27 + 20 + 33 + 12 + 4 + 6 + 8 = 147
9 + 5 + 15 + 16 + 1 + 31 + 21 + 32 + 17 = 147
9 + 11 + 3 + 23 + 13 + 19 + 14 + 29 + 26 = 147
9 + 25 + 24 + 10 + 22 + 7 + 30 + 2 + 18 = 147
4. If we select four numbers from eight arranged on each circle, their sum equals the magic number, 69. By forming a semi-circle that contains these numbers, we can create eight such semi-circles.
20 + 33 + 12 + 4 = 69
6 + 8 + 28 + 27 = 69
32 + 17 + 5 + 15 = 69
16 + 1 + 31 + 21 = 69
23 + 13 + 19 + 14 = 69
29 + 26 + 11 + 3 = 69
10 + 22 + 7 + 30 = 69
2 + 18 + 25 + 24 = 69
Below are a few alternatives to the given magic circle. Similar results hold for these representations as well. Readers are encouraged to verify all of them.
Example – II
Let us now consider another example of a magic circle of order 6. In this instance, we use 49 natural numbers (1, 2, 3, …, 49) without any repetition. The task is to fill these circles with the natural numbers from 1 to 49 (without repetition) such that the sum of the numbers along every line is the same.
This example was devised by Ding Yidong, a mathematician contemporary with Yang Hui. The magic circle of order 6 is presented below:
The following facts are apparent from the figure:
1. The sum of the numbers on the four diameters is 325. There are four diameters, and the four sums are, respectively, as follows.
1 + 11 + 21 + 31 + 41 + 5 + 25 + 45 + 9 + 19 + 29 + 39 + 49 = 325
2 + 12 + 22 + 32 + 42 + 10 + 25 + 40 + 8 + 18 + 28 + 38 + 48 = 325
3 + 13 + 23 + 33 + 43 + 15 + 25 + 35 + 7 + 17 + 27 + 37 + 47 = 325
4 + 14 + 24 + 34 + 44 + 20 + 25 + 30 + 6 + 16 + 26 + 36 + 46 = 325
2. The sum of 8 numbers at the outermost circle added to 9 gives at the centre gives 300:
1 + 46 + 47 + 2 + 49 + 4 + 3 + 48 = 200
11 + 36 + 37 + 12 + 39 + 14 + 13 + 38 = 200
21 + 26 + 27 + 22 + 29 + 24 + 23 + 28 = 200
31 + 16 + 17 + 32 + 19 + 34 + 33 + 18 = 200
41 + 6 + 7 + 42 + 9 + 44 + 43 + 8 = 200
5 + 30 + 35 + 10 + 45 + 20 + 15 + 40 = 200
3. The sum of the numbers positioned diametrically opposite is the same. For instance:
1 + 49 = 50
11 + 39 = 50
21 + 29 = 50
31 +19 = 50
41 + 9 = 50
5 + 45 = 50
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