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 2 of 5: Determine the only possible group sizes

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q=\left\lfloor\frac{2n}{17}\right\rfloor,\qquad\text{every group has size }q\text{ or }q+1

Detailed analysis

There are $2n$ people in $17$ groups. If two allowed sizes differ by at most one, their sizes must be the floor and ceiling of the average $2n/17$ . Thus, with $q=\lfloor2n/17\rfloor$ , every group has size $q$ or $q+1$ .