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?

(In reply to meaning by Solomon)

The statements made regarding "triangular" were not in reference to the digits, but to the four-digit number [7] and the number representing the sum of the digits [8].  (As to single digits, 1, 3, and 6 are each triangular numbers, yet neither the four-digit number nor the number of its digit sum is a single digit).

To quote the formal definition stated in Mathworld, "a triangular number is a number obtained by adding all positive integers less than or equal to a given positive integer n".

You may find a list of the first few trianglular numbers in The On-Line Encyclopedia of Integer Sequences (Sloane's A000217 ).

Edited on March 18, 2013, 12:05 am

Posted by Dej Mar on 2008-05-28 12:56:54