In a group of students, 50 speak English, 50 speak French and 50 speak
Spanish. Some students speak more than one language. Prove it is
possible to divide the students into 5 groups (not necessarily equal),
so that in each group 10 speak English, 10 speak French and 10 speak
Spanish.
Let EF denote the number of students speaking English and French. Similarly define ES, FS, E, F, S, EFS. Then ES + EF + E + EFS = 50, EF + FS + F + EFS = 50. Subtracting: ES - F = FS - E. Similarly, ES - F = EF - S.
Pair off members of FS with members of E. Similarly, members of ES with F,and members of EF with S. The resulting pairs have one person speaking each language. If ES = F, then the only remaining students are those in EFS, who speak all three languages. We thus have a collection of units (pairs or individuals) each containing one speaker of each language.
If ES < F, then after the pairing off we are left with equal numbers of members of E, F, and S. These may be formed into triplets, with each triplet containing one speaker of each language. As before we also have the students in EFS. Again, we have partitioned the student body into units with each unit containing one speaker of each language.
If ES > F, then after the pairing off, we are left with an equal number of members of ES, FS and EF. These may be formed into triplets, with each triplet containing two speakers of each language. So, in this case we partition the student body into units with each unit containing either one speaker of each language, or two speakers of each language.
Finally, we may divide the units into 5 groups with 10 speakers of each language in each group.
Puzzle Solution
