ch. x] kirkman's school-girls problem 213 



First arrange these residues in pairs so that every difference 

 between the numbers in a pair occurs once. One rule, by which 

 this can be effected, is to divide the residues into two equal 

 sections and pair the numbers in the two sections. This gives 

 (2. 11), (4. 9), (8. 5), (3. 10), (6. 7), (12. 1) as possible pairs. 

 Another such rule is to divide the residue into six equal 

 sections, and pair the numbers in the first and second sections, 

 those in the third and fourth sections, and those in the fifth 

 and sixth sections. This gives (2. 8), (4. 3), (6. 11), (12. 9), 

 (5. 7), (10. 1) as possible pairs. Either arrangement can be 

 used, but the first set of pairs leads only to scalene triangles. 

 In none of the pairs of the latter set does the sum of the 

 numbers in a pair add up to 13, and since this may allow the 

 formation of isosceles as well as of scalene triangles, and thus 

 increase the variety of the resulting solutions, I will use the 

 latter set of pairs. We use these basic pairs as suffixes of the 

 ' a 's, and each pair thus determines two points of one of the 

 triangles required. We have now used up all the ' a'a. The 

 third point associated with each of these six pairs of points 

 must be a ' b,' and the remaining six ' b 's must be such that 

 they can be arranged in suitable triplets. 



Next, then, we must arrange the 6u residues el, el, e3, ... in 

 possible triplets. To do this arrange them cyclically in triplets, 

 for instance, as shown in the first column of the left half of the 

 annexed table. We write in the second column the differences 

 between the first and second numbers in each triplet, in the 

 third column the differences between the second and third 

 numbers in each triplet, and in the fourth column the differences 

 between the third and first numbers in each triplet. If any 

 of these differences d is greater than 3m we may replace it 

 by the complementary number y — d: that this is permissible 

 is obvious from the geometrical representation. By shifting 

 cyclically the symbols in any vertical line in the first column we 

 change these differences. We can, however, in this way always 

 displace the second and third vertical lines in the first column 

 so that the numbers in the second, third, and fourth columns 

 include the numbers 1 to 3u twice over. This can be effected 



