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PROBLEMS AND SOLUTIONS. 
571 
41 
Solution by the Proposer. 
Let the numbers be 1, 2, 3, 4, 5, 6, 7, 8, 9. Any number, say 1, enters into three 
triads, no two of which have any number in common. We may take these triads to 
be 123, 145, 167. There remain the two numbers 8, 9; and these are, or are not, a 
duad of the system. 
First Case.—8 and 9 a duad. In the triad which contains 89, the remaining 
number cannot be 1; it must therefore be one of the numbers 2, 3; 4, 5 ; 6, 7; and 
it is quite immaterial which; the triad ma}^ therefore be taken to be 289. There is 
one other triad containing 2, the remaining two numbers thereof being taken from the 
numbers 4, 5; 6, 7. They cannot be 4, 5 or 6, 7; and it is indifferent whether they 
are taken to be 4, 6; 4, 7; 5, 6, or 5, 7: the triad is taken to be 247. We have 
thus the triads 
123, 145, 167, 289, 247; 
and we require two triads containing 8 and two triads containing 9. These must be 
made up with the numbers 3, 4, 5, 6, 7: but as no one of them can contain 47, it 
follows that, of the two pairs which contain 8 and 9 respectively, one pair must be 
made up with 3, 5, 6, 7, and the other pair with 3, 5, 6, 4; say, the pairs which 
contain 8 are made up with 3, 5, 6, 7, and those which contain 9 are made up with 
3, 5, 6, 4 (since obviously no distinct case would arise by the interchange of the 
numbers 8, 9). The triads which contain 8 must contain each of the numbers 
3, 5, 6, 7, and they cannot be 835, 867, since we have 67 in the triad 167; similarly 
the triads which contain 9 must contain each of the numbers 3, 5, 6, 4, and they 
cannot be 845, 836, since we have 45 in 145. Hence the triads can only be 
836, 857 
837, 856 
934, 956, 
935, 946 ; 
and clearly the top row of 8 must combine with the top row of 9, and the bottom 
row of 8 with the bottom row of 9; that is, the system of the nine triads is 
123, 145, 167, 289, 247, 
in combination with 
or else in combination with 
836, 857, 934, 956, 
837, 856, 935, 946. 
These are really systems of the same form, that is, each of them is of the form 
viz. in the first and second systems respectively we have 
ABCafiyabc 
6 1 3287549 (First system), 
5 1 3294678 (Second system),