60 THE MAGIC SQUARE. 



3) The numbers from i 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. 



i 



25 54 



31 49 



10 15 80 



76 36 44 9 



50 70 72 32 



55 

 5 45 ^ 6 5 37 



11 6I 60 24 x 



3 20 17 53 



4 56 59 43 



3* . 21 64 4 3 



69 73 



6 62 57 58 23 79 



75 67 



77 8 



41 19 63 l8 38 



46 . 33 



12 39 68 74 42 13 

 51 28 



4 29 34 7 78 47 52 3 

 Fig. 35. 



4) The numbers from i 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 



