IX.

MAGICAL POLYGONS.

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 ]