# Sunday Times Teaser 2581 – Knowing The Lingo

*by RBJT Allenby*

#### Published May 14 2017 (link)

A group of 64 students can speak, between them, French, German and Russian. There are three times as many French speakers and two times as many German speakers as there are Russian speakers. The number who speak all three languages is one fifth of the number who speak French and at least one other of these languages. The number who speak all three languages is also two ninths of the number who speak German and at least one other of these languages. Furthermore, the number who speak all three languages is also two fifths of the number who speak Russian and at least one other of these languages.

How many of the 64 students speak only French?

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This is most easily visualized using a Venn diagram as follows:

where the circles represent the students who speak French, German and Russian respectively and their overlapping segments represent those students who speak different language combinations; the outer circle represents those who speak none. The text in the eight areas will be used to denote each of the subgroups using a subscript on the symbol \(n\) giving the number of students in each of the eight sub-groups.

We can now write down six equations among eight variables for the conditions set in the teaser as follows:\[n_n + n_f + n_g + n_r + n_{fg} + n_{fr} + n_{gr} + n_{fgr} = 64\]\[n_f + n_{fg} + n_{fr} + n_{fgr} = 3(n_r + n_{fr} + n_{gr} + n_{fgr})\]\[n_g + n_{fg} + n_{gr} + n_{fgr} = 2(n_r + n_{fr} + n_{gr} + n_{fgr})\]\[n_{fgr}= (1/5)(n_{fr} + n_{fg} + n_{fgr})\]\[n_{fgr}= (2/9)(n_{fg} + n_{gr} + n_{fgr})\]\[n_{fgr}= (2/5)(n_{fr} + n_{gr} + n_{fgr})\]

The last three of these equations can be easily manipulated to give:\[n_{fgr}=2n_{gr}\]\[n_{fr}=2n_{gr}\]\[n_{fg}=6n_{gr}\]and these allow the other equations to be reduced to:\[n_f=3n_r + 5n_{gr}\]\[n_g=2n_r+n_{gr}\]\[n_n+n_f+n_g+n_r+11n_{gr}=64\]and finally:\[n_n+6n_r+17n_{gr}=64\]

This is a Frobenius Equation for which my number theory library has a solver.

which gives the following 28 solutions:

This table gives the numbers in each of the eight sub-groups followed by the number of students who speak French, German and Russian. The teaser text doesn’t allow the last of these and it seems certain that the intended solution is the first in which all students speak at least one of the three languages.