The math teacher addressed the class. "I have chosen a positive integer (base 10). It has 4 digits, in ascending sequence, none of which are 0. I have calculated the sum of its digits, the sum of the squares of its digits, and the product of its digits, and each of these quantities has a different number of digits. Consider the following 9 statements:
(1) The number is a prime.
(2) The sum of its digits is a prime.
(3) The sum of the squares of its digits is a prime.
(4) The number is a square.
(5) The sum of its digits is a square.
(6) The sum of the squares of its digits is a square.
(7) The number is triangular.
(8) The sum of its digits is triangular.
(9) The sum of the squares of its digits is triangular.

You must use them to determine the number."
"But they can't all be true!", interjected one of the students.
"I never said they were! Some of the statements are true and some are false."
"Well we will need more information. Tell us which are true and which are false."
"If I told you that you would easily be able to determine the number!"
"Well at least tell us how many are true."
"If I told you that now, you would be able to determine the number too easily!"
"Well, what more can you tell us?"
"Nothing! I have told you enough!"

What was the number?

Does "4 digits, in ascending sequence" mean that the digits are strictly increasing? If I'm not mistaken, something 1111 is considered to be ascending.

Originally, I narrowed down the numbers (including repeating digits) and then literally checked all the requirements. Unless I did something wrong or left out something, I could not find a unique solution.

I then eliminated possibilities with repeating digits, then there was a unique solution. It sure would've saved me a lot of trouble if I knew this in advance.

So please clarify.

Posted by np_rt on 2004-12-23 19:48:38