April 3, 1902] 



NA TURE 



509 



LETTERS TO THE EDITOR. 

 ( The Editor does itot hold hiniseif responsible for opinions eX' 

 pressed by his correspondents. Neither can he undertake 

 to retitrn, or to cot respond ivith the writers of, rejei/ei 

 manuscripts intended for this or any other part of NATURE. 

 No notice is taken of anonviioits communicat ions.'\ 



Magic Squares of the Fifth Order. 



In the interesting discourse reproduced in your issue of 

 March 13 (p. 447), there is a statement that the number of magic 

 squares of order 5 exceeds 60,000. Major MacMahon informs 

 me that he gave these figures on the authority of Rouse Ball's 

 " Mathematical Recreations." The statement is not wrong, 

 but viewed as a minimum limit it may be largely exceeded. I 

 have recently investigated the total number of squares of this 

 order, which have the additional property that the nine 

 numbers in the heart of the square also form a m.igic — the 

 well-known " bordered squares." Fig. i is an example. The 

 square itself is magic in rows, columns and diagonals, and the 

 nine numbers in the central square show like properties. 



It is easy to see (i) that the numbers in the central square 

 must consist of three arithmetical progressions with a like 

 common difference ; 12) that the first terms of each progression 

 must also be in arithmetical progression ; and (3) that the mean 

 number (13) must always occupy the central cell. Hence it 

 follows that (excluding thf^ central number) the eight numbers 

 in the heart must consist of four pairs of complementary 

 numbers (i.e., pairs whose sum is 26, or twice the mean), and 

 also that the two smallest numbers being known, then all the 

 others are known. If, for example, the two lowest numbers in 

 the heart are a, a^k, then the nine numbers in the centre, 

 taken in numerical order, must be a, a + h, a + 2k ; i^-h, I3i 

 1 3 + i ; 26 - 2h - a, 26 - Jt - a, 26 -a. It is now easy to deter- 

 mine in how many ways we may choose the nine numbers for 



the centre on the usual assumption that we are restricted to the 

 first 25 natural numbers without repetition or omission. The 

 total is 26, as shown in the first column of the table below, 

 where the figures in brackets denote the two lowest numbers in 

 the heart. 

 Type. 



A. (I, 2) 



B. (I, 3) 

 C (I, 4) 



D. (I, 6) 



E. (2, 3) 



F. (2, 4) 



G. (2. 5) 

 H. (2, 6) 

 I- (2,7) 

 I- (3.4) 

 K. (3, 5) 

 L. (3, 6) 

 M. (3,7) 



Number c 

 subtypes, 



... 30 



Total 



602 



These we may call the 26 main types of bordered squares of 

 the fifth order. The 16 remaining numbers have evidently to 

 be arranged in the four borders in such a way that comple- 

 mentary pairs are opposed at the opposite ends of rows, 

 columns, or diagonals, with the additional condition that the 

 four borders must each sum 65. Now when we deal a 

 pair of complementary numbers into the top and bottom rows, 



NO. 1692, VOL. 65] 



we clearly give to those rows a difference equal to the difference 

 of the complementary pair. If we call this a " complementary 

 difference," then the numbers in the two rows must be so 

 related that the sum of two complementary differences must 

 equal the sum of the other three, and a similar relation must 

 hold between the two lateral borders. Suppose we are dealing 

 with type L. (3, 6). The numbers for the centre are 3, 6, 9 ; 

 10, 13, 16 ; 17, 20, 23. If now we arrange the remaining 

 numbers in two columns, with complementaries adjacent, and 

 form a third column of half differences, we have : — 



It is then necessary to form two equations from the column of 

 half-differences of the (oiir. a + b=c + d + e, and so related that 

 two and only two numbers shall be common to both equa- 

 tions, and that these two shall be on opposite sides of the 

 equality sign in one equation, and on the same side in the 

 other. In the case of type L. we can do this in thirteen 

 different ways, as shown below, the italicised figures denoting 

 the corner pairs : — 



12 + 11= g + S + 6 \ 

 6+ 5= <S' + 2+i J '^• 



12+ 6= 9 + S+\} 

 II + 5= S+6-\-2J ■ 



12 + ^"= 9 + 6 + 5), 

 >S'+6=ii+2+iJ '■ 



12 + 6= 9 + 8 + /\ , 

 9 + 5 = 11+2 + // • 



12+5= 9 + 6 + 2\f 



5 + 6=ii+3+iy- 

 12 + /= 6+5 + 2) , 



9 + 8= n+j- +-//•*• 

 II +9 = 12 + 6 + 2) . 



6 + 5= 8+-.'+iF- 



// + 8=I2+J- + 2) , 

 //+j-= g + 6+l/ • 



y III. 

 12+2 + 1 ) 



Each of these 13 pairs yield a subtype under type L. For 

 example, take the first pair. The first equation tells us that the 

 greater of the two pairs of numbers whose half-differences are 12 

 and 1 1 can be associated in one border with the lesser of the 

 three pairs whose half-differences are 9, S and 6, the comple- 

 ments, of course, being opposed in the opposite border. The 

 other equation gives similar information regarding the other two 

 opposed borders. The centre can be arranged at once by follow- 

 ing the order of the ordinary magic of the third degree. The 

 result is shown in Fig. 2. 



The table above gives the number of subtypes for each main 

 type, the total being 602. The result has been verified by 

 another worker independently. 



It is now easy to calculate how many bordered magics of the 

 fifth order exist. The centre for a given subtype we know 

 can be arranged in one way only if reversions and reflections are 

 not reckoned as different ; in eight ways if they are so reckoned. 

 Consider the top row in any one of the 602 subtypes. For the 

 left-hand corner we have four choices, for the next cell six, and 

 for the third cell two. This leaves three choices for the second 

 cell of the left-hand border and two for the third. The 

 numbers in all the remaining border cells are now known. 

 There are, therefore, 4x6x2x3x2 = 28S ways of arranging 

 the borders for each sub-type. The total number of squares is 

 thus 602 X 288 X 8 if reversions and reflections are reckoned as 

 different ; 602 x 288= 173,376 if they are not so reckoned. 



When we bear in mind that this is the number of magics of 

 a restricted type, it is clear that the number of magics of the 

 fifth order must largely exceed this total ; indeed, everything 

 suggests that the totality of magics of the fifth order is more than 

 twice as great as the above result. C. Planck. 



Ilaywards Heath, March 15. 



