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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?

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

1 Answer 1

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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 $a+b=22$).

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