The Nasik square (so named by the Rev. A. H. Frost after the town in India where he lived) is the most perfect type conceiv able. Here all the broken diagonals sum to the constant 34. Thus, for example, 15+14+2+3 and 10+4+7+13 and 15+5+2+12. As a consequence its properties are such that if you repeat the square in all directions you may mark off a square 4 X4 wherever you please and it will be magic. It should be noted that Nasik squares can be constructed for any doubly-even orders and for any odd orders not divisible by 3. But a Nasik square of a singly even order is impossible. The number of fundamentally different Nasik squares of the 5th order is sixteen.
In the case of the associated squares every number if added to the number that is equidistant in a straight line from the centre gives 17. Thus 1+16, 2+15, 3+14, etc. Diirer's square (fig. 2) is also associated The simple square is one that fulfils the simple conditions and no more. The semi-Nasik has the property that the opposite short diagonals of two cells together sum to 34. Thus 14+4+11+5=34. 12+6+13+3=34. And in this order there are 48 Nasiks. 384 semi-Nasik (which include 48 associated) and 448 simple, making a total of 880. The Nasik square is also sometimes called diabolique and pandiagonal and the semi-Nasik semi-diabolique.
A graphic illustration of each type is shown in figs. i i to 14, if placed beneath the squares to which they apply. Every Nasik square takes the form of fig. every associated that of fig. 12, but the graphic forms for the simple and semi-Nasik squares are con siderably varied. There are 12 such graphic forms in all of the 4th order, and these are shown in Amusements in Mathematics. (See bibliography.) Another type of magic square is the bor dered, an example of which is shown in fig.
15 where it will be seen that a square of the 3rd order is surrounded by a border so as to form a square of the 5th order. In fig. 16 we have an example of an extension first considered by Frenicle (c. 1602-75). Here the borders may be successively stripped off to produce magics of the 9th, 7th, 5th and 3rd orders, and these concentric or progressive squares for odd and even orders respectively may be constructed without any limit whatever.
Composite squares also may be formed in certain cases. If we know how to construct a square of the mth and nth orders we can directly make one of the math order as in fig. 17 where m and a are each 3. It will be noted that each subsidiary square is succes
sively constructed in the same order as the smallest one and each successive square is placed in the larger square, again in the same order. A more difficult but very elegant type of square is that of two degrees, as in fig. 18. Here we have a magic square of the 8th order with the constant 26o and every square of f our cells sum ming to 130. If for every number we substitute in its allotted place its square then it will also be entirely magic with the constant I Ii80. A Nasik square of this kind and order has been constructed.
was given by Ernest Bergholt in Nature for March 26, 1910. Then the question of the construction of magic squares with prime num bers only, and with the smallest possible constants, has been inves tigated. A summary of results will be found in The Monist (Chi cago) for Oct. 1913. The formation of squares composed of con secutive composite numbers has also been considered. These can be formed for any order without the use of tables of primes though the method will not always give the smallest possible constant. First write down any consecutive numbers, the smallest being greater than 1, say, for the 3rd order, 3, 4, 5, 6, 7, 8, 9, '0. The only prime factors of these numbers are 2, 3, 5 and 7. Add No square of two degrees is possible for any order smaller than the 8th.