12 



♦ KNOWLEDGE ♦ 



[November 1, 1886. 



Calling the several giils a, b, c, d, &c., write down all 

 the combiiiatious of the set, in pairs, in order thus : — 



AB, AC, AD, AE, AF, &C. 



EC, BD, BE, BF, &C. 



CD, CE, CF, &C. 



DE, DF, <fcc. 



EF, &C. 



&C. 



Then repeating the first row, write under it in succession 

 such of the pairs of the second, third, fourth, &c., rows in 

 succession as can walk with the pairs in the first row. Do 

 the like with the pairs in the third and other rows, always 

 putting in the next available pair on a row in the first 

 available place, and filling up according to tlie requirements 

 for each day's walk, that Ls, so that every girl may go out 

 each day. The result is bound to come out right. 



Thus, suppose there are eight girls. A, b, c, d, e, f, g, h, 

 then the lists come out as follows : — 



1st day. 2nd day. 3rd day. 4th day. 5th day. 6th day. 7tli day. 



AB AC AD AE AF AG AH 



CD BD BE BF EG BH BC 



EG EH CP CG CH CE DG 



EH EG GH DH DE DF EF 



But although this method always solves the problem satis- 

 factorily, there is a more systematic plan worth knowing. It 

 was given by Mephisto in Vol. III. of Knowledge (No. 71). 

 The solution for the case we have just been dealing with is 

 shown in the following figure : — 



Here the letters represent the eight girls, the seven days 

 on which they are to walk are indicated by the numbers, 

 and a number on any row and column signifies that on the 

 day bearing that number the two persons represented by the 

 two letters marking that row and that column are to walk 

 together. Thus those squares running diagonally across the 

 figure which mark severally a row and a column belonging 

 to the same person remain, of course, unnumbered. The 

 numbers on the first row and column are 1,2, 3, 4, 5, 6, 

 and 7, showing that A is to walk on the seven successive 

 days with b, c, d, e, f, g, and H. Next, according to 

 Mephisto's way of presenting the method, write in along 

 the second low and column the numbers in order from 1 

 (at the left and top respectively), omitting the square which 

 is to be left vacant, and setting the number 2, which would 

 have fallen on that square, on the last row and the right- 

 band column respectively. We do the same with the third 

 row and column, setting the omitted figure, 4, over to the 



bottom and to the right ; and .so on to the end. We thus 

 complete a schedule, which would arrange the girls as 

 follows : — 



1st day. 2nd day. 3rd daj'. 4th day. 5th day. Gthday. 7th day. 



The 7-aiionale of this method is obvious. Omitting the h row 

 and the h column, and supposing the vacant squares occu- 

 pied by the numljers left out in the actual solution, we see 

 that the successive rows may be supposed to be obtained by 

 shifting the pi'eceding row one square to the left and carry- 

 ing the square thi'ust out in this way on the left over to the 

 right. This, of course, ensures that every number will fall 

 in succession opposite each letter both on I'ow and on column, 

 so that, applying the result, each person of the .set, whether 

 of schoolgirls or of chess-players, will be paired against every 

 other in succession. The same result may be obtained more 

 conveniently, however, as follows : — 



Here we have only seven letters, but interpreting the 

 numbers as in the other method, we see that on certain 

 days we get aa, bb, cc, &c. Introducing in each case an 

 eighth letter h, for the second member of these doubles, and 

 taking the columns in order from left to right, we get the 

 following arrangement : — ■ 



the same arrangement as before. This plan is not only 

 easier and simpler, but it serves better, I think, to indicate 

 the principle on which the method depends. We see that 

 each number representing a day must come in four times, 

 and must get a different ])air of letters for each time it 

 appears. Take, for example, the Gth day, and begin with 

 the 6 in the first column, which gives us the pair af ; the 

 next 6 being one to the right and one above takes the pair 

 BE ; the next takes the pair CD ; and we see that the 

 remaining 6 corresponds to gg, and so gives the pair gh. 

 We can also see why the number which in the other way of 

 working this method would fall on the square to be left 

 vacant is carried over to column and row belonging to the 

 eighth letter. 



My plan is also convenient because the same figures may 

 be used for all cases. Suppose the first seven lines above 

 only shown, and that the first column is continued by 

 running on the letters in order, the other columns by 



