# Phone numbers and lovers

One day, a person was in love. He wanted to find the person's phone number.

So they start up a conversation.

Person: What's your phone number, sweetheart?

Person's Love: I'm not going to give it that easily. If you replace the first digit of your phone number with the lowest odd digit, you'll get my number.

Person: And the 4-digit code before it?

Person's Love: You know, the product of the four digits in the code is actually the square root of my phone number.

Person: Hey, that's insufficient. I'm a plumber, not God.

Person's Love: But if I tell you the sum of digits of the code, I'd have told you too much.

Person: I love you, dear.

Person's Love: I love you too.

So what is the 4-digit code?

The information given is that:

• The product of the digits is $X$, where $X$ is a number that the person knows, but that we do not know. $X^2$ starts with an odd digit.
• The person does not know the number given only the product, but if the sum of the digits was given, then the person would know the number.

The first thing to notice is that

If the digits of the code are rearranged, the product and sum stay the same. So all four digits must be the same. Otherwise there would be multiple possible codes and the second bullet point would be false.

Given that, the possible values of $X$ are

The fourth powers of each digit: $0$,$1$,$16$,$81$,$256$,$625$,$1296$,$2401$,$4096$, or $6561$.

We can rule out the values whose squares start with an even digit, leaving

$1$,$625$,$1296$, and $4096$.

But if $X$ was

$1$, $625$, or $4096$

then there would only be one possible code that produced that product, and the person have known already. So we can eliminate these possibilities. This means $X$ must be

$1296$

and the code must be

$6666$

• Beware! The beast~ – Quiquȅ Feb 22 '16 at 2:29