Tom, Dick and Harry are three brothers, Hearty, hale and youthful;
And each of them is always lying, Or – is always truthful.

‘Most of them are truthful, though,’ Claimed their doting mother.
So I went to ask the lads To tell about each other.

Then Tom declared that Dick denied, That Harry always lied.
‘Tom tells a lie, I tell you so,’ Brother Harry cried.

I am confused, I must confess, And now I turn to you.
Can you tell me who was lying, And who is always true?

^{– Marta Sved}

“I’m just wilde about Harry!”

Hint 1

Consider assigning a letter T (truthful) or L (liar) to each of the brothers.

In how many ways can you do this? How many of these cases are disallowed by the mother’s claim?

Hint 2

Use the other information given. Try some cases (as above), and see if you obtain a contradiction.

Solution

There are just eight ways of assigning the letters T or F to the brothers. From the mother’s comment, the only allowable rows must have at least two Ts. Hence only the last four rows are possible.

Let us now analyze the statement: Harry says: ‘Tom tells a lie.’” If Harry tells the truth, then Tom is lying. If Harry is lying, then Tom is truthful. Hence rows 6 and 8 of the table are impossible. This leaves us with rows 5 and 7 as possibilities.

Suppose row 5 gives the answer: i.e. suppose Tom and Dick are truthful and Harry is lying. Then Tom’s truthful (if rather complicated!) statement is that Dick (truthfully) denied that Harry always lies. That is, Dick (truthfully) claims that Harry tells the truth.

Extensions

There are a great number of puzzles about people who may or may not be telling the truth.

The interested reader may care to refer to the books by Raymond Smullyan: for example, What is the name of this book? (Pelican 1981), and The lady or the tiger ((Pelican 1982).

Hint 1

Hint 2

Solution

Extensions