IMC 2016 Q22
22. In a particular group of people, some always tell the truth, the rest always lie. There are 2016 in the group. One day, the group is sitting in a circle. Each person in the group says, “Both the person on my left and the person on my right are liars.”
What is the difference between the largest and smallest number of people who could be telling the truth?
- A 0
- B 72
- C 126
- D 288
- E 336
Intermediate Mathematical Challenge 2016, Question 22. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 336
For brevity, let T denote a truth teller and L a liar. Clearly each T has to have an L on each side. Each L either (i) has a T on each side or (ii) has an L on one side and a T on the other side. The largest number of Ts will occur if (i) is always the case. This gives the arrangement TLTLTL… which, since 2 divides 2016, joins up correctly after going round the table. In this case the number of Ts is \(\frac{1}{2} \times 2016\). The smallest will occur if case (ii) always is the case. This gives the arrangement LLTLLTLLT… which, since 3 divides 2016, also joins up correctly. In this case the number of Ts is \(\frac{1}{3} \times 2016\). The difference is \(\frac{1}{6} \times 2016 = 336\).
Solution © UK Mathematics Trust, from the IMC 2016 solutions. Longer versions with investigations: Solutions and investigations (PDF).