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 4 of 5: Enumerate the admissible values

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9\le n\le16;\ 18\le n\le24;\ 27\le n\le32;\ 36\le n\le40;\ 45\le n\le48;\ 54\le n\le56;\ n\in\{63,64,72\}

Detailed analysis

Substitute $y=17-x$ , $9\le x\le16$ , and $q=\lfloor2n/17\rfloor$ into the four inequalities. Checking the resulting finite ranges gives exactly $9$ – $16$ , $18$ – $24$ , $27$ – $32$ , $36$ – $40$ , $45$ – $48$ , $54$ – $56$ , $63,64,72$ (all are at most $1996$ ). No other $n$ satisfies the necessary inequalities.