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?
Best Answer
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 $a+b=22$).