So far we have only considered such extensions of the idea underlying
the construction of the magic square in which the figure of the square
was retained. We may however contrive extensions of the idea in which
instead of a square, a rectangle, a triangle, or a pentagon, and the
like, appear. Without entering into the consideration of the methods for
the construction of such figures, we will give here of magical polygons
simply a few examples, all supplied by Professor Scheffler:
1) The numbers from 1 to 32 admit of being written in a rectangle of 4 ×
8 in such a manner that the long horizontal rows give the sum of 132 and
the short vertical rows the sum of 66; thus:
[Illustration: Fig. 31.
+--+--+--+--+--+--+--+--+
| 1|10|11|29|28|19|18|16|
+--+--+--+--+--+--+--+--+
| 9| 2|30|12|20|27| 7|25|
+--+--+--+--+--+--+--+--+
|24|31| 3|21|13| 6|26| 8|
+--+--+--+--+--+--+--+--+
|32|23|22| 4| 5|14|15|17|
+--+--+--+--+--+--+--+--+
]
2) The numbers from 1 to 27 admit of being so arranged in three regular
triangles about a point which forms a common centre, that each side of
the outermost triangle will present 6 numbers of the total summation 96
and each side of the middle triangle 4 numbers whose sum is 61; as the
following figure shows:
[Illustration: Fig. 32.
26 3 6 10 24 27
20 9 11 21
18 2
16 17
15 8
22 5
12
7 13
4 23
19
1 14
25
]
3) The numbers from 1 to 80 admit of being formed about a point as common
centre into 4 pentagons, such that each side of the first pentagon from
within contains two numbers, each side of the second pentagon four
numbers, each of the third six numbers, and each side of the fourth,
outermost pentagon eight numbers. The sum of the numbers of each side of
the second pentagon is 122, the sum of those of each side of the third
pentagon is 248, and that of those of each side of the fourth pentagon
254. Furthermore, the sum of any four corner numbers lying in the same
straight line with the centre, is also the same; namely, 92.
[Illustration: Fig. 33.
1
26 54
31 49
10 15 80
76 36 44 9
50 70 72 32
55 71 16 66 27
5 45 25 65 37 2
11 61 60 24 14
30 20 17 53
40 56 59 43
35 21 64 48
69 57 58 73
6 79
77 75 62 23 67 8
46 41 19 22 63 18 38 33
51 12 39 68 74 42 13 28
4 29 34 7 78 47 52 3
]
4) The numbers from 1 to 73 admit of being arranged about a centre,
in which the number 37 is written, into three hexagons which contain
respectively 3, 5, and 7 numbers in each side and possess the following
pretty properties. Each hexagon always gives the same sum, not only when
the summation is made along its six sides, but also when it is made along
the six diameters that join its corners and along the six that are
constructed at right angles to its sides; this sum, for the first hexagon
from within, is 111, for the second 185, and for the third 259.
[Illustration: Fig. 34.
1 5 6 70 60 59 58
63 8
62 19 53 46 22 45 9
61 20 24 64
2 48 31 42 38 49 57
3 47 39 40 44 56
67 51 41 37 33 23 7
66 50 34 35 54 11
65 25 36 32 43 26 12
10 30 27 13
17 29 21 28 52 55 72
18 71
16 69 68 4 14 15 73
]