Problem 4

The National Marriage Council wishes to invite

n

couples to form

17

discussion groups. Each group contains members of only one sex; the sizes of any two groups differ by

0

1

; every group is nonempty; and every person belongs to exactly one group. Find all

n\le1996

for which this is possible.

Step 5 of 5: Construct groups for every listed value

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r_m=n-xq,\quad r_f=n-yq,\quad 0\le r_m\le x,\quad0\le r_f\le y

Detailed analysis

For each listed $n$ , the inequalities provide an $x$ and $y$ . Make $r_m=n-xq$ male groups of size $q+1$ and $x-r_m$ of size $q$ ; make $r_f=n-yq$ female groups of size $q+1$ and $y-r_f$ of size $q$ . The bounds ensure the remainders lie in the displayed ranges, so all conditions hold. This proves sufficiency.