# Question from the 2011 IMC (international mathematics competition) key stage III paper, about a logical sequence

There are two kinds of students in a certain class, those who always lie and those who never lie. Each student know what kind each of the other students is. In a meeting today, each student tells what kind each of the other students is. The answer "liar" is given $$240$$ times. Yesterday a similar meeting took place, but one of the students did not attend. The answer "liar" was given $$216$$ times then. How many students are present today?

The question above, was posed, in the 2011 international mathematics competition. I have been studying, for quite a while, but to no avail. I am valiantly attempting, to create a logical connection, between the lies and the truths, without success. Can you guys please guide me, create a logical connection between the two and eventually, work out the answer, to this problem?

• Not sure what IMC this is from, but it did appear on AoPS in 2014. Feb 11, 2020 at 18:23
• Consider bipartite graph of truthers (knights) and liars. On each of edges "liar" sounds $2$ times. Thus $xy=120$, $\,x(y-1)=108$. $120=2^3\cdot 3\cdot 5$, $\,108=2^2\cdot 3^3$ $\implies$ $x=12,\,y=10$, $x+y=\color{red}{22}$ Jul 1, 2020 at 23:07

In a group of $$a$$ liars and $$b$$ truth-tellers, each truth-teller will identify $$a$$ liars and each liar will "identify" $$b$$ liars. Hence in total, there will be $$2ab$$ answers "liar". Today, we have $$2ab=240$$ and yesterday, we had $$2(a-1)b=216\qquad\text{or}\qquad 2a(b-1)=216.$$ By subtracting we get $$2b=240-216=24$$ (and thereby $$a=10$$, $$a+b=22$$) or $$2a=24$$ (and thereby $$b=10$$ and again $$a+b=22$$).