John is tryng to locate Tom's house. All he knows is that Tom lives on a street with houses numbered from 8 to 100.

John asks Tom:

"Is your house number bigger than 50?"

Tom answers him, but he lies. (John doesn't know that he's lying) John continues to ask:

"Is your house number a multiple of 4?"

Tom answers and again, he lies. Then John asks:

"Is your house number the square of an integer?"

Tom answres, but this time he tells the truth. Finally, John asks:

"Is the first digit of your house number 3?"

After Tom answers (we don't know if he lied or not) John declares Tom's house number, but he is wrong! What is Tom's house number?

81.

The first answer Tom gives is, "no." This is false, but John believes it is true, as is evidenced by his later asking if the number begins with 3.

The second answer Tom gives is "yes." This is false, but John believes it is true. I will explain this in a moment.

The third answer Tom gives is "yes." This is the truth. Tom now believes the number one of the following: 4, 16, 36, given that these are the only squares that are multiples of 4 below 50.

The fourth answer Tom gives is "yes." This is a lie, but John believes it is true, therefore he guessed 36, but was wrong.

The way we can be certain that the answers Tom gave are those above is by looking at John's process of elimination. If Tom had said "no" to question 2, John would not have asked the fourth question, as no squares that are NOT multiples of four start with 3. A yes answer to Q2, Q3 and Q4 are the only answers that would allow John to give a single answer with certainty. Any deviation from those answers would leave an ambiguous answer after the fourth question.

Since we have established the answers to the questions, we can analyze them and their veracity to determine the actual number:

Q1: answer - no, veracity - false
50 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100
Q3: answer - yes, veracity - true
X = 81 (the only square above 50 that is not a multiple of four)

The fourth question is required to establish an answer for John, but we do not need it.

Please answer if there is a flaw in my logic.

Posted by Freshter on 2002-05-31 22:12:11