Perimeter Magic Polygon >k=3

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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

Perimeter Magic Squares

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.

[1] Terrel Trotter, Jr., Normal Magic Triangles of Order n, Journal of Recreational  Mathematics, Vol. 5,, No. 1, 1972, pp.28-32
[2] Terrel Trotter, Jr., Perimeter-magic Polygons, Journal of Recreational  Mathematics, Vol. 7,, No. 1, 1974, pp.14-20
[3] For information on Perimeter magic cubes see my unusual cubes page.

Perimeter Magic Pentagons

These are the only order-3 perimeter magic pentagons (not counting the 4 rotations and 5 reflections of each).
By computer search, I have found 6074 solutions for order-4. 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 Bi-magic order-4 pentagon

On the previous page we saw a Perimeter Bi-magic order-4 triangle.

Can someone find an Perimeter Bi-magicorder-4 square or hexagon?

Perimeter Magic Hexagons

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 order-4 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,
13 to 30, 31 to 54, and 55 to 84.

 

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 7-sided and 9-sided polygons here.

Then for higher odd orders, these can be modified by adding pairs of numbers to each side (as per discussion under Pentagons).

 


 

 

This page was originally posted October 2006
It was last updated August 13, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright 1998-2009 by Harvey D. Heinz