In 1972, Terrel Trotter, Jr.,
then a math teacher in Urbana Illinois, published an article called Normal
Magic Triangles of Order n [1]. In 1974, he published a follow up article
called Perimetermagic Polygons [2]. In it he introduced the term
Perimetermagic Polygon,
and generalized it to include other
polygonal shapes. Since that time, other writers have expanded on the same
theme.
On this page, I will attempt to
expand on, and summarize the published material on perimetermagic
polygons (PMPs for short) and to add interest, I will include some
problems dealing with the subject matter. Because of the nature of
perimetermagic polygons, these problems will be suitable for middle
school students. A companion page will feature magic graphs. That page
also will include problems. However, these will tend to be much more
complicated and will, in many cases, require the use of a personal
computer.
Additional pages on this subject
Order 3
Perimeter Magic Triangles
Order3 is the simplest perimeter
magic triangle, except for order 1 which is trivial (there are no order 2
perimetermagic objects). Unlike magic squares, cubes, etc., there are
multiple magic constants for the same order.

Here I show the only PMPs of order 3.
Solution 4 is the compliment of solution 1, and solution
3 is the complement of solution 2
(see below). 
Complements.
For any given solution, there is always a complement solution. It is obtained by
subtracting each number in turn from the sum of the first and last numbers in
the series. So if your solution uses the numbers from 1 to 9, another solution
is obtained by subtracting each number from 10 (ie. The sum of the first and
last numbers in the series). This feature applies to magic squares, cubes etc.
as well.
The new solution obtained in this way
must then be normalized by rotation and/or reflection, to comply with the
above
For higher orders, with many solutions,
it is necessary to only find solutions for the smallest half of the possible
sums. The higher magic sums may then be obtained simply by complementing these
lower solutions.
Rotations and Reflections

Rotations and Reflections are not
considered unique solutions. Each basic solution has two rotations and
each of these has one (horizontal) reflection, for a total of 6 apparently
different solutions. For other polygons, these numbers would obviously
vary. While any one of these six could be considered the basic
solution, I have chosen to use the convention described in the next
section. 
Standard for Normalized
solutions
There are many ways to sort magic
perimeter solutions.
Other writers have chosen to sort by side or vertex totals (or possibly no sort
at all).
For polygons with few solutions, this step is not important. However, as the
number of solutions increases, a sorted list is necessary in order to prevent
possible duplications.
I have chosen to use the following
system.
To facilitate putting polygon solutions in order:
Place the lowest vertex number at the top
Then move clockwise around the polygon and
Enter vertex numbers in order, then
Enter side numbers in order
Arrange list of solutions in sorted order
Order 3 solutions in list
order:
In the sorted list below, S=Side
sum (i.e. the magic constant, V=Vertex sum, C=the complement solution #, Bold
columns are vertices
# A B C S V C
1. 1 6 2 4 3 5 9 6 4
2. 1 6 3 2 5 4 10 9 3
3. 2 5 4 1 6 3 11 12 2
4. 4 3 5 1 6 2 12 15 1
Note that solutions 3 and 4 have been
normalized before being added to the list
An example of complementing and normalizing:
# A B C S V C
1. 1 6 2 4 3 5 9 6 4 Solution 1
4. 6 1 5 3 4 2 Subtract above number from 7
4. 4 3 5 1 6 2 12 15 1 Normalized solution 4
Some Math
Formulas required for solving
perimetermagic polygons.
Trotter covered the mathematic involved in
great detail. Here I am only summarizing.
We are assuming the numbers used are
consecutive starting with 1.
The following notation will be used.
k = the number of
sides of the polygon
n = the number of
integers per side (the order)
S = the magic constant (Trotter used C, but S is consistent with magic square
terminology)
N = the number of
consecutive integers being used
Vertex sum = k[S1/2(n1){(n1)k+1}]
The smallest vertex sum is 1 + 2 + 3
i.e. the 3 smallest integers in the series from 1 to N.
The largest vertex sum is 3n – 3 (that is, the 3 largest integers).
However, because of parity, the 3 smallest integers and the 3 largest integers
may not always give the possible smallest and largest vertex sums. There are two
separate cases, depending on the values of n and k:
If n is even or n and k
are both odd, then
min V_{s}_{ }= ½k(k +
1) max V_{s}_{ }= ½k[(2n – 3)k
+ 1]
When n is odd and k is even, then
min V_{s}_{ }= ½k(k +
2) max V_{s}_{ }= ½(2n3)k^{2}
The final values we need to determine
are the minimum and maximum values of the magic constant S.
Here again we must have two formulas, depending on the values of n and
k.
If n is even or n and k
are both odd, then
min S = (½n^{2} – n + 1)k + ½ n)
max S = (½n^{2} – 1)k + ½ n
When n is odd and k is even, then
min S = (½n^{2} – n + 1)k + ½ (n
+ 1) max S = (½n^{2} – 1)k + ½ (n –
1)
A word of caution. These formulas
indicate where solutions may be found. On very rare occasions, there may
be no solution for a particular value of S. However, only two such cases are
known. They are; for a 4^{th} order triangle with S = 18 or 22, and, 3^{rd}
order pentagon with S = 15 or 18.
For perimeter magic triangles, a simple
equation gives the number of magic sums for any given order.
Number of different sums = 3n5. This even allows for the special ‘case
of the two missing sums in the order 4 triangle
Can you find any
other cases with no solution for a value of S between the theoretical minimum
and maximum limits?
If there are any such cases, they will always be
an even number because of the complement feature mentioned above.
A table generated from the above
formulae has been placed at the beginning of each polygon section following.
[1]
Terrel Trotter, Jr., Normal Magic Triangles of Order n,
Journal of Recreational Mathematics, Vol. 5,, No. 1, 1972, pp.2832
[2] Terrel Trotter, Jr., Perimetermagic Polygons, Journal of
Recreational Mathematics, Vol. 7,, No. 1, 1974, pp.1420
A guide to finding solutions
I find the simplest method for
constructing PMP’s manually is the following:
k =
the number of sides of the polygon
n = the number of
integers per side (the order)
Vs = the sum of the vertex numbers
Ns = sum of the series of numbers used
Ms = the average midside sum (ie sum of integers between the two
vertices of each side.
 First choose the integers for the
vertices from the set N where N = (n1)k
 The magic sum S where S
= (Ns+Vs)/n (this must be an integer if a valid solution
is possible).
 Find the average midside sum where
Ms = (NsVs)/n
 Using the above step as a guide,
partition the remaining numbers in the series into n groups with the
required sums to place on the sides of the polygon (this should always be
possible if the result of step 2 was an integer.
Order 4
Perimeter Magic Triangles
There are 18 basic solutions for
order 4 PM Triangles. Note that there are no solutions with magic sums of
18 and 22.
The number of solutions for each sum should be symmetrical.
Sum 17 19 20 21 23
# of solutions 2 4 6 4 2 = 18
The list below contains all solutions
for order 4 PM Triangles.
S=Side sum (i.e. the magic constant,
V=Vertex sum, C=the complement solution #,
CS=the complement side sum, CV=the complement vertex sum. Columns A, B, and C
are the vertices
# A B C S V C
1. 1 5 9 2 4 8 3 6 7 17 6 17
2. 1 5 9 4 2 6 7 3 8 19 12 12
3. 1 6 8 2 5 7 3 4 9 17 6 18
4. 1 6 8 4 3 5 7 2 9 19 12 14
5. 1 6 8 5 2 4 9 7 3 20 15 6
6. 2 4 9 5 1 6 8 3 7 20 15 5
7. 2 5 9 3 1 8 7 4 6 19 12 10
8. 2 6 7 5 3 4 8 1 9 20 15 8
9. 2 6 8 3 4 5 7 1 9 19 12 13
10. 3 2 9 7 1 5 8 4 6 21 18 7
11. 3 4 8 5 2 6 7 1 9 20 15 11
12. 3 4 8 6 1 5 9 2 7 21 18 2
13. 3 5 6 7 2 4 8 1 9 21 18 9
14. 3 5 7 6 2 4 9 1 8 21 18 4
15. 4 2 9 5 1 8 6 3 7 20 15 15
16. 4 3 8 5 2 7 6 1 9 20 15 16
17. 7 2 6 8 1 5 9 3 4 23 24 1
18. 7 3 5 8 2 4 9 1 6 23 24 3
Some example diagrams:
Solution # 17 is the normalized
complement of solution #1, and # 12 is the normalized complement of #2.
Number 8 is it’s own complement.

Solution number 6 is of special
interest because it is bimagic. It was originally discovered by David
Collison of California, USA [3]. When each number is squared, the triangle
is still perimeter magic. This time with a sum of 126.
This suggest another
investigation…to find bimagic solutions in other orders (or even trimagic
solutions?). 
[3] John R.
Hendricks, The Magic Square Course, 2^{nd} edition, 1992, 522
pages +,
Larger
Perimeter Magic Triangles
Order5 Triangles
Order 5 PM Traingles have ten values for
S. They range from 28 to 37. Trotter [1] reports that there are 1356 basic (he
calls them primitive) solutions.
Here I show solutions for the 4 lowest
possible magic sums. By complementing these, you can obtain another solutions
for the 4 highest sums. I leave it to you to find solutions for sums of 32 and
33.
Note that I have not supplied solution numbers to these. The only way that can
be done is to find and list in order all of the possible basic solutions.
Orders6 to 8
Triangles
Here is summary information on orders 6
to 8 perimeter magic triangles. For completeness, I included the same
information for the smaller triangles.

Order 3 
Order 4 
Order 5 
Order 6 
Order 7 
Order 8 
Minimum S 
9 
17 
28 
42 
59 
79 
Maximum S 
12 
23 * 
37 
64 
74 
97 
Integers used 
1 to 6 
1 to 9 
1 to 12 
1 to 15 
1 to 18 
1 to 21 
Minimum Vertex sum 
6 
6 
6 
6 
6 
6 
Maximum Vertex sum 
15 
24 
33 
42 
51 
60 
Number of basic solutions 
4 
18 
1356? 
? 
? 
? 
* For order
4, there are no solutions with sums of 18 or 22.
Following are sample solutions of
orders 6, 7, and 8 PM Triangles. The reader is left with a challenge to find
additional solutions. Or even, all the solutions for a given order.
Perimeter magic triagonals are neither
as complex, or mathematically challenging as magic squares or cubes.
Nevertheless, they can be of interest to anyone who is intrigued with the vast
variety of number patterns.
Other Types of
PM Triangles
Subtractive magic triangles
Sunday Ajose
[4] proposed a type of triangle where the
smallest number on each side is subtracted from the sum of the two largest
numbers. For orders greater then 3, the sum of the smaller numbers is subtracted
from the sum of the larger numbers on each side. In all cases, to be magic the
differences must be the same. He calls this constant d. Some examples
follow.
Some questions posed by the author:
Can a subtractive magic triangle be formed from the first 3n natural numbers for
any order > 3?
Does a subtractive magic triangle retain it’s magic property if each element is
increased by a constant?
Does a subtractive magic triangle retain it’s magic property if each element is
multiplied by a constant?
What is the maximum and minimum differences obtainable for subtractive magic
triangles of a given order?
[4] Sunday A. Amose, Subtractive Magic Triangles, Mathematics Teacher, 76
(5), 1983, pp 346347
Balanced perimeter triangles
This pattern was suggested by Charles
Trigg [5].
The digits 1 to 9 may be partitions into
3 groups, each summing to 15. We can place these groups onto the 3 sides of a
triangle, but with no numbers at the vertices. Thus each side will sum to 15.
There is one group consisting of 2, 2 and 5 integers, eight groups of 3, 3, and
3 integers, and 6 groups of 2, 3, and 4 integers.
I leave it as an exercise for the
reader, to find the partitions for the first and last set. Also, how many
different unique solutions are possible (don’t count rotations and reflections)?
The eight partitions for 3 sets of 3 are
elegantly represented by the 3 x 3 magic square. The main diagonals cannot be
used because the share the common number 5. The three rows may be placed on the
3 sides of a triangle for one arrangement. The three columns may be placed on
the sides of a triangle for another solution.
Any position is permitted for the numbers on a side, so there are a lot of
possible solutions.

Figure 1 is the set represented by
the rows of the magic square. Figure 2 is represented by the columns.
While this is the simple way to determine which side to place the numbers,
there are two other methods (I know of) to do so.
Place the numbers on the sides of
the triangle in sequence, but at the end of each triad, skip one side (fig
1), or at the end of each triad, place the next number on the same side
(fig 2). 
For added interest, this problem may be
generalized for polygons of any number of sides.
[5] Charles W. Trigg, Triangles With Balanced Perimeters, Journal of
Recreational Mathematics, Vol. 3, No. 4, 1970, pp 255256
Antimagic triangular arrays.
This digression from perimeter magic
triangle, explores another article by Charles Trigg
[6]. He investigates formations of numbers
arranged in triangular arrays. However, for order 3, these numbers do appear on
the perimeter of a triangle (on higher orders, there are interior number as
well).

Here is one such array were the
sums of each side are in consecutive order (ie. 11, 12, 13). As a bonus,
the order 2 triangles sums are 8, 9, 10. The inverted order 2, consisting
of the 3 midside numbers, sum to 6, different but not in consecutive
order with the others.
A few minutes of doodling produced
the order 4 perimeter antimagic triangle shown. The order 4 side sums are
an arithmetic progression order (13, 20, 27). The three order 2 triangles
also sum in arithmetic progression order (12, 15, 18). In an order 4
perimeter triangle there is no inverted center triangle. 
In keeping
with magic square terminology, these triangles should more properly be called
heterotriangles because the sums are different but not in consecutive order. For
more indepth information, see my antimagic squares page [7]
[6] Charles W. Trigg, Special Antimagic Triangle Arrays, Journal of
Recreational Mathematics, Vol. 14, No. 4, 198182, pp 274278
[7]
For information on antimagic squares see
my antimagic squares
page
[1] Terrel Trotter, Jr., Normal Magic Triangles of Order n,
Journal of Recreational Mathematics, Vol. 5,, No. 1, 1972, pp.2832
[2] Terrel Trotter, Jr., Perimetermagic Polygons, Journal of
Recreational Mathematics, Vol. 7,, No. 1, 1974, pp.1420
A scan of this article is at
http://www.trottermath.net/simpleops/pmp.html
[3] John R. Hendricks, The Magic Square Course, 2^{nd} edition,
1992, 522 pages +,
[4] Sunday A. Amose, Subtractive Magic Triangles, Mathematics Teacher, 76
(5), 1983, pp 346347
[5] Charles W. Trigg, Triangles With Balanced Perimeters, Journal of
Recreational Mathematics, Vol. 3, No. 4, 1970, pp 255256
[6] Charles W. Trigg, Special Antimagic Triangle Arrays, Journal of
Recreational Mathematics, Vol. 14, No. 4, 198182, pp 274278
[7]
For information on antimagic squares see
my antimagic squares
page
Other related
references:
Yates, Daniel S., Magic Triangles and a Teachers Discovery, Arithmetic
Teacher 23, 1976, pp 35154.
Lowell A. Carmony, A Minimathematical Problem: The Magic Triangles of Yates,
Mathematics Teacher, 70 (5), 1977, pp. 410413
Janet Caldwell, Magic Triangles, Mathematics Teacher, 71 (1), 1978, pp.
3942
Peggy A. House, More Mathemagic from a Triangle, Mathematics Teacher, 73
(3), 1980, pp. 191195
Charles W. Trigg , Second Order Perimeter–magic and Perimeterantimagic
Cubes, Mathematics Magazine, 47 (3), 1974, pp9597
Charles W. Trigg , Edge Magic and Edge Antimagic Tetrahedrons, Journal of
Recreational Mathematics, Vol. 4, No. 4, 1971, pp 253259
Charles W. Trigg , Edge Magic Tetrahedrons with Rotating Triads, Journal
of Recreational Mathematics, Vol. 5, No. 1, 1972, pp 4042
Charles W. Trigg, Eight Digits on a Cubes Vertices, Journal of
Recreational Mathematics, Vol. 7, No. 1, 1974, pp 4975
Charles W. Trigg, Perimeter Antimagic tetrahedrons and Octahedrons,
Journal of Recreational Mathematics, Vol. 11, No. 2, 197879, pp 105107
For information on Perimeter magic cubes see my
unusual cubes page.
