How to Solve a Magic Hexagon with Formula Properties and Solved Examples
Magic hexagon of order n is an array 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 of magic hexagon sum to the same magic constant M. A normal magic hexagon includes the consecutive integer from 1 to 3n² - 3n + 1, whereas the abnormal magic hexagon starts with the number other than 1. It is concluded that normal magic hexagons exist only for n = 1 (which is trivial) or n = 3. The first magic hexagon that was introduced has a magic sum of 1 and the second magic hexagon has a sum of 38.
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The numbers in any row of the above hexagon with order n = 3 sums to 38. For example, 3 + 17 + 18 = 38, 19 + 7 + 1 + 11 = 38, 12 + 4 + 8 + 14 = 38.
A magic hexagon for trigonometric identities is a special diagram that helps you to remember trigonometric identities. Here, we look at how the magic hexagon for trigonometry helps to remember different trigonometric identities.
Magic Hexagon For Trigonometric Identities
Trigonometric identities are equalities that include trigonometric functions and are true for every value of the variables that occur for which both the sides of equalities are defined. The trigonometric identities are useful whenever expressions including trigonometric functions are required to be simplified.
The magic hexagon is a special diagram that helps you to quickly memorize different trigonometric identities such as Pythagorean, reciprocal, product/function, and cofunction identities. Also, you will learn how trigonometric identities are useful to evaluate trigonometric functions.
Building Magic Hexagon For Trigonometric Identities
Construct the hexagon and locate ‘I’ at the center of the hexagon.
Write tan on the farthest left vertex of the magic hexagon as shown in the figure given below.
Use quotient identities ( tan x = sin x/cos x ) for the tangent going clockwise as shown in the figure below.
Locate the “co” - functions such as cot (cotangent), csc (cosecant), and sec (secant) on the opposite vertex of the hexagon as shown in the figure below.
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To help you remember: all the “co” functions are placed on the right side of the hexagon.
Along the outside edges of the hexagon, we can now follow around the clock (in either direction) to get the quotient identities. The quotient identities given below are in two equivalent forms of each.
Quotient Identities
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The two trigonometric functions located on any diagonals of a hexagon are reciprocal of each other.
Reciprocal Identities
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Product Identities
The magic hexagon given here shows that the trigonometric function between two functions is equal to them and multiplied together. If the identities are opposite to each other, then 1 is between them. Check the product identities below to know better.
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Cofunction Identities
The trigonometric functions such as cosine, cotangent, and cosecant on the right side of the hexagon are the cofunction of the trigonometric functions that are on the left side of the hexagon such as sine, tangent, and secant. Hence, sine and cosine are conjunctions. Let us learn to form the cofunction identities with the help of the figure given below.
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Examples:
The value of sine (30°) = cos (60°)
The value of tan (80°) = cos (10°)
The value of sec (40°) = csc (50°)
Cofunction Identities in Radians
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Examples:
The value of sine (0.1π) = cos (0.4 π)
The value of tan (π/4) = cot (π/4)
The value of sec (π/3) = csc (π/6)
The Pythagorean Identities
The unit circle shows that sin² x + cos² x = 1
The magic hexagon also helps us to remember Pythagorean identities by moving clockwise, around any of the triangles below.
For each shaded triangle in the figure below, the addition of the upper left function squared and upper right function squared obtains the bottom function square.
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We can write Pythagorean Identities as:
Solved Examples
1. Which of the following Quotient Identities given below is accurate?
Sin y = Sin (y)/Cot (y)
Sin y = Cos (y)/Cot (y)
Sin y = Sec (y)/Tan ( y)
Sin y = Cos (y)/Sec ( y)
Solution:
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The magic hexagon shows that the identity Sin y = Cos(y)/Cot(y) is correct.
2. Which of the following reciprocal identities is accurate?
Cot y = 1/Tan (y)
Cot y = 1/Sin (y)
Cot y = 1/Sec (y)
Cot y = 1/Csc (y)
Solution:
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The magic hexagon shows that the identity Cot y = 1/Tan(y) is accurate.
Abnormal Magic Hexagons
Even though normal magic hexagons with orders of more than 3 are not known, certain “abnormal” magic hexagons are known to exist. In this case, the word "abnormal" refers to the state where the starting of the sequence of numbers is other than with 1. Such abnormal hexagons were first discovered by Arsen Zahray, who discovered the order 4 and 5 magic hexagons
We can understand this by the following facts: the order 4 hexagon starts with 3 and ends with 39, and the sum of its row to 111. Similarly, an order of 5 hexagons will begin with 15, end with 75, and add up to 305.
A total higher than 305 for an order 5 hexagon is not possible.
In a magic hexagon of 5th order, "X" will refer to the placeholders for order 3 hexagons, which complete the number sequence. In the upper fits the hexagon gives a total sum of 38 (numbers 1 to 19) while in the lower one of the 26 hexagons will have the sum equal to 0 (numbers -9 to 9).
An order of six hexagons starts with 21, ends with 111, and its sum is 546. It was designed by mathematician Louis Hoelbling on October 11, 2004.
And the magic hexagon of order 7 was discovered by Arsen Zahray with the help of annealing simulation method on 22 March 2006:
A seventh-order magic hexagon starts with 2, ends with 128, and sums up to the value of 635.
Similarly, the magic hexagon of the order 8 was created by Louis K. Hoelbling on February 5, 2006, and it begins with -84 and ends with 84, and its sum is 0.
Magic T- hexagons
It is interesting to know that magic hexagons can also be constructed with triangles and this type of configuration is referred to as a T-hexagon. It has several more properties than the hexagon of hexagons.
In general, a T-hexagon of order n has 6n2 triangles. The sum of all these numbers is given by: S = 3n2(6n2+1)
In order to make a magic T-hexagon of side n, 'n' must be even, because there are r = 2n rows so the sum in each row must be
M= S/r
There must be an even number of 'n' for 'M' to be an integer. To date, magic T-hexagons only of the order 2, 4, 6, and 8 have been discovered. The first magic hexagon that was a magic T-hexagon of order 2 was supposedly discovered by John Baker on 13 September 2003. Since then, John has had several collaborations with David King (who is well known for the discovery that there are 59,674,527 non-congruent magic T-hexagons of 2nd order).
Magic T-hexagons are sometimes thought to be similar in a number of ways to magic squares, but they have their own special features.
FAQs on Magic Hexagon in Mathematics Explained Clearly
1. What is a magic hexagon in mathematics?
A magic hexagon is a hexagonal arrangement of consecutive numbers where all rows in three directions add up to the same total, called the magic sum. In a standard order-3 magic hexagon, the numbers 1 to 19 are placed so that:
- Each straight line (horizontal and the two diagonal directions) has the same sum.
- The common total is the magic constant.
- It is similar in concept to a magic square but arranged in a hexagonal pattern.
2. What is the magic sum of a magic hexagon using numbers 1 to 19?
The magic sum of the standard magic hexagon using numbers 1 to 19 is 38. This is calculated by:
- Adding all numbers from 1 to 19: 1 + 2 + ... + 19 = 190.
- In an order-3 magic hexagon, each of the 15 lines must total the same value.
- The only arrangement that works gives a constant sum of 38 for every straight line.
3. How do you solve a magic hexagon step by step?
To solve a magic hexagon, you must arrange consecutive numbers so every straight line equals the magic sum. Basic steps include:
- Step 1: Determine the total sum of all numbers.
- Step 2: Calculate the required magic constant.
- Step 3: Place middle or central numbers strategically (usually medium values).
- Step 4: Adjust outer numbers to balance all three directional sums.
4. Is there more than one solution to the magic hexagon?
There is essentially one unique solution to the standard order-3 magic hexagon, ignoring rotations and reflections. This means:
- All valid solutions are symmetrical versions of the same arrangement.
- No completely different number pattern satisfies the conditions.
5. Why is the number 10 placed in the center of a magic hexagon?
The number 10 is placed at the center because it is the median of the numbers 1 to 19 and balances the magic sums. Since:
- The total sum is 190.
- The central position belongs to multiple lines.
- A middle value ensures symmetry and equal distribution.
6. What is the formula for the sum of numbers in a magic hexagon?
The total sum of consecutive numbers in a magic hexagon is found using the formula n(n + 1) / 2, where n is the largest number. For example:
- For numbers 1 to 19:
- Total sum = 19 × 20 / 2 = 190.
7. What is the difference between a magic hexagon and a magic square?
The main difference is that a magic square uses a square grid, while a magic hexagon uses a hexagonal arrangement with three directional sums. Key differences:
- Magic square: rows, columns, and diagonals sum equally.
- Magic hexagon: three sets of parallel lines sum equally.
- Magic hexagon is geometrically more complex.
8. Can you give an example of a magic hexagon arrangement?
One valid magic hexagon arrangement uses the numbers 1–19 so each line sums to 38. A simplified structure (row-wise) is:
- Row 1: 3 numbers
- Row 2: 4 numbers
- Row 3: 5 numbers
- Row 4: 4 numbers
- Row 5: 3 numbers
9. Are larger magic hexagons possible?
A normal magic hexagon with consecutive numbers exists only for order 3. Mathematicians have proven that:
- No other size works using consecutive integers starting from 1.
- Order-3 (19 cells) is the only possible standard solution.
10. What are common mistakes when solving a magic hexagon?
A common mistake when solving a magic hexagon puzzle is failing to balance all three directional sums simultaneously. Typical errors include:
- Ignoring diagonal line totals.
- Placing very large or very small numbers in the center.
- Forgetting the required magic sum is 38 (for 1–19).
