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 1 of 5: Count male and female groups

Previous step Next step

x+y=17,\qquad x\ge y,\qquad n\ge x\ge9,\qquad 1\le y\le8

Detailed analysis

Let $x$ and $y$ be the numbers of male and female groups. Both sexes occur, so $x,y\ge1$ ; exchange the names if necessary to assume $x\ge y$ . Since $x+y=17$ , we have $9\le x\le16$ and $1\le y\le8$ . Every male group is nonempty, so $n\ge x\ge9$ .