Look at it this way:
Label the people from 1 to 9 then continue with letters A to N.
Pair them in the following way:
13 24
35 46
57 68
79 8A
9B AC
BD CE
DF EG
FH GI
HJ IK
JL KM
LN M2
N1
23 pairings with each label occurring exactly twice.
This set of pairings is not unique of course.