On this page I will present some condensed information on perimeter magic polygons of sides 4, 5, 6, etc. I will also present a brief description on a simple construction method.
Perimeter Magic Squares  Perimeter Magic Pentagons 
Perimeter Magic Hexagons  Constructing Odd Order PMPs 
This table shows relevant information for 4 sided PMPs.
Squares 
Order 3 
Order 4 
Order 5 
Order 6 
Order 7 
Order 8 
Minimum S 
12 
22 
37 
55 
78 
104 
Maximum S 
15 
30 
48 
71 
97 
128 
Integers used 
8 
12 
16 
20 
24 
28 
Minimum Vertex sum 
12 
10 
12 
10 
12 
10 
Maximum Vertex sum 
24 
42 
56 
74 
88 
106 
Number of basic solutions 
6 
146? 




* These table are calculated using the formulae in [1] and [2]
These are the six possible solutions for order 3. They are in sorted order with 1 and 6, 2 and 5, 3 and 4 the complement pairs.
Order 4 has a total of 146 basic solutions (I think). Here are four.
Some examples for higher orders of perimeter magic squares.
For information on Perimeter magic cubes see my http://members.shaw.ca/hdhcubes/cube_unusual.htm#Perimetermagic
These are the only order3 perimeter magic pentagons (not
counting the 4 rotations and 5 reflections of each).
By computer search, I have found 6074 solutions for order4. However, I have not
confirmed that that is all the possible basic solutions.
These six solutions are shown arranged in complement pairs,
This table shows relevant information for 5 sided PMPs.
Pentagons 
Order 3 
Order 4 
Order 5 
Order 6 
Order 7 
Order 8 
Minimum S 
14 
27 
45 
68 
96 
129 
Maximum S 
19 
37 
60 
88 
121 
159 
Integers used 
10 
15 
20 
25 
30 
35 
Minimum Vertex sum 
15 
15 
15 
15 
15 
15 
Maximum Vertex sum 
40 
65 
90 
115 
140 
165 
Number of basic solutions 
6 
6074? 




The following four perimeter magic pentagrams were constructed with the help of a simple routine (once the smallest odd and even orders are designed. Divide the extra numbers required into pairs with equal sums. Then add one of these pairs to each side of the original PMP to get the next larger order of the same parity.
In this case, to obtain the order 6 PM pentagon, partition the extra numbers (16 to 25) into 5 pairs each totalling 41. Then add one of these pairs of numbers to any side of the originating order 4.
A Perimeter Bimagic order4 pentagon On the previous page we saw a Perimeter Bimagic order4 triangle. Can someone find an Perimeter Bimagicorder4 square or hexagon? 
This table shows relevant information for 6 sided PMPs.
Hexagons 
Order 3 
Order 4 
Order 5 
Order 6 
Order 7 
Order 8 
Minimum S 
17 
32 
54 
81 
115 
154 
Maximum S 
22 
44 
71 
105 
144 
190 
Integers used 
12 
18 
24 
30 
36 
42 
Minimum Vertex sum 
24 
21 
24 
21 
24 
21 
Maximum Vertex sum 
54 
93 
126 
165 
198 
237 
Number of basic solutions 
20 





The 20 basic solutions for order 3 Hexagons
# A B C D E F S Comp. # 1 1 11 5 10 2 12 3 8 6 4 7 9 17 19 2 1 11 5 9 3 12 2 8 7 4 6 10 17 20 3 1 11 5 8 4 10 3 12 2 6 9 7 17 18 4 1 12 6 2 11 5 3 9 7 4 8 10 19 14 5 1 11 7 9 3 4 12 2 5 6 8 10 19 7 6 1 9 8 6 4 12 2 11 5 3 10 7 18 17 7 1 11 8 7 5 3 12 2 6 4 10 9 20 5 8 1 10 8 4 7 9 3 11 5 2 12 6 19 10 9 1 10 8 2 9 6 4 12 3 5 11 7 19 15 10 1 11 8 2 10 4 6 9 5 3 12 7 20 8 11 2 12 6 4 10 1 9 3 8 5 7 11 20 16 12 2 12 6 3 11 5 4 7 9 1 10 8 20 13 13 2 10 7 1 11 5 3 12 4 6 9 8 19 12 14 2 11 7 1 12 3 5 9 6 4 10 8 20 4 15 2 8 10 1 9 7 4 11 5 3 12 6 20 9 16 3 12 4 10 5 8 6 2 11 1 7 9 19 11 17 3 10 8 2 11 1 9 7 5 4 12 6 21 6 18 4 7 11 1 10 3 9 5 8 2 12 6 22 3 19 6 9 7 5 10 1 11 3 8 2 12 4 22 1 20 6 9 7 3 12 2 8 4 10 1 11 5 22 2
Three examples
Three order 4 perimeter magic hexagons with consecutive vertex numbers. All solutions for order4 PM Hexagons have not yet been compiled (to my knowledge), so we cannot assign solution numbers to these figures.
This illustration combines 4 orders of perimeter magic hexagons consisting of consecutive numbers. It is an example of the fact that like magic squares and cubes, these figures also remain magic when a constant is added to each number in the series. In this case, the resulting series
are 1 to 12,

Constructing Odd
Order PMPs
A simple procedure for constructing order 3 perimeter magic polygons is illustrated here. The numbers may be written down in consecutive order as per the sequences shown in the 7sided and 9sided polygons here. Then for higher odd orders, these can be modified by adding pairs of numbers to each side (as per discussion under Pentagons). 
[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
[3] For information on Perimeter magic cubes see my
http://members.shaw.ca/hdhcubes/cube_unusual.htm#Perimetermagic
.Please send me Feedback about my Web site!
Harvey Heinz harveyheinz@shaw.ca
Last updated October 19, 2006
Copyright © 1999 by Harvey D. Heinz