Truth tellers and Liars

Here is famous puzzle pattern, I wish to put in the following way.

Assume three friends A, B and C are sitting on a concrete wall. The friends always tell either truth or always lie, and they knew each other who tell truth. One person, asked A, “are you a truth teller”? A said something but the person could not able to listen.

The person asked B, what did A said? He replied “A says he is always truth teller”.

Again, the person asked C, what did A said? He replied, “B is lying”.

Question is, C – truth teller or liar?

Hint: The answer is obvious. We need to transform these statements into mathematics. There is a branch of mathematics called connectives. We can make use of them to arrive at solution.

The connectives are AND (conjunction), OR (disjunction), Conditional (if then) and Bi-conditional (if and only if). Make truth tables for each and relate them with statements.

We can make many of such puzzles based on the connectives properties.

Find the ratio of ages

The following puzzle I posted to one of Orkut polls, try it.

Assume X age is K years now. X is twice as old as Y was when X was as old as Y is now. Find the age ratio of X and Y

Hint: Let A and B are age of X and Y respectively, after C years it will be such that A + C = k (B + C), where k is integer.

The answer is 4:3. How is it possible? My emphasis is on problem solving approach, not the answer.

The number 1729

The blog I started to share few of mathematical techniques and puzzles that I learned. It is honor to start with a puzzle discovered by the genius Ramanujan.

In 1917, Ramanujan became seriously ill. He was incorrectly diagnosed with tuberculosis; later, it was confirmed vitamin deficiency. When Ramanujan was sick, Prof. Hardy visited him. Hardy told him that the number of the cab he came in, 1729, was a "dull number" and he hoped that it wasn't a bad omen. "No, sir," Ramanujan replied. "It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways". How would you find them?

Hint: Use your personal computer and basic C language constructs. How the genius did ensured that the number was smallest under the mentioned conditions?

Some one coined, every prime number is personal friend of Ramanajun.