An interesting math problem

Today I want to share an interesting math problem . This problem is from Math Kangaroo.The problem is as follows.

Each of the 2017 people living on an island is either a liar(and always lies) or a truth-teller(and always tells the truth). More than one thousand of them take part in a banquet, all sitting together at a round table. Each of them says: “ Of the two people beside me, one is a liar and the other one a truth-teller.” What is the maximum number of truth-tellers on the island?

(A)1683 (B) 668 (C) 670 (D)1344 (E)1343

What is your answer?

Let me show you how to solve it in my way.

The first step in solving this problem is to find how many people sitting at the round table.If we want to find the number of the people sitting at table ,we should determine what pattern the people sitting at the table .Since everyone says “ Of the two people beside me, one is a liar and the other one a truth-teller.” Assuming that the person who says this is a liar, because he always lies, so “ Of the two people beside me, one is a liar and the other one a truth-teller” is a lie.That means either two liars or two truth-teller sitting beside him.And the possible seating arrangements are

(1) liar, liar, liar, liar, liar, liar, …

(2) truth-teller, liar, truth-teller, truth-teller, liar ,truth-teller, truth-teller, liar,truth-teller,...

In the first case, all of them can only be liars. If so, the number of truth-tellers cannot reach the maximum number. In the second case, it can be seen that two thirds of people are truth-tellers. It says that more than 1,000 people attended the banquet. To maximize the number of people who tell the truth,we should find the smallest number which is over 1,000 and multiple of 3.That’s 1002.Therefore ,two thirds of 1002 people are sitting at the round table and the rest of 2017 who are not sitting at the table are all truth-teller.Two-thirds of 1002 is 668, 2017-1002+668=1683. The answer is (A).

We can also assume that the person who says, “ Of the two people beside me, one is a liar and the other one a truth-teller.” is truth-teller, which means “ Of the two people beside me, one is a liar and the other one a truth-teller.” is true. Then there is a liar on his one side and a truth-teller on his other side and the possible seating arrangement is

liar,truth-teller,truth-teller, liar,truth-teller,,truth-teller, liar,truth-teller,truth-teller ...

In this patter,we can see that two out of three people must be the ones who tell the truth. Then answer should be A ,too.

This is my solution to the problem.

How do you find your answer?

Do you get it right?