# Happy New Year

Pro Problems > Math > Number and Quantity > Number Theory > Digits## Happy New Year

Happy New Year! I am a four-digit year, and my last two digits are a perfect square. The sum of my first and third digits is a perfect square. My second digit is a perfect square. All my digits add to a perfect square.

If you subtract my first, second, and third digit from my last digit, you get a perfect square.

If you subtract my third digit from my first digit, you get a perfect square.

Oh, by the way, I'm a perfect square.

What year am I?

## Solution

In order to make it feasible for teachers to use these problems in their classwork, no solutions are publicly visible, so students cannot simply look up the answers. If you would like to view the solutions to these problems, you must have a Virtual Classroom subscription.## Similar Problems

### Three Digit Difference

Two positive integers, A and B, both have 3 digits. A is bigger than B. A – B is between 300 and 400. What is the value of A - B?

### Back to Back

X is a three-digit number. Y is the number obtained when the digits of X are reversed. Z is the six-digit number obtained by writing X and Y back to back, with X written first. W is the six-digit number obtained by writing Y and X back to back, with Y written first. What is the largest number which the sum of Z and W must be divisible by?

### Two Digit Pattern Matching

How many two-digit numbers are there such that the digits match at least one of the following patterns:

- The digits are both multiples of three.
- Neither of the digits are multiples of two.
- The digits add to 8.
- The digits are perfect squares.

### Rhonda's Zip Code

Rhonda’s zip code has five digits. Two of the digits are the same. One of the digits is three times another digit. Three of the digits are consecutive integers. The zip code starts with a zero. What is the largest possible sum for the digits of Rhonda’s zip code?

### Three Digit Number

I am thinking of a three-digit number. The sum of my digits is 17. Two of my digits add to 10, and two of my digits are the same. Find all possible values for my number.

### Four Digit Number

I am a four digit number.

The sum of my digits is 20.

The product of my digits is 600.

The difference between my first two digits is 2, and the sum of my middle two digits is 11.

What number am I?

### Sum of Digits

Find the sum of all the integers between one and 100 which have 14 as the sum of their digits.

### Set of Five Digit Numbers

S is the set of five-digit numbers such that the digits are in ascending order, there are no repeated digits, the sum of the first two digits is equal to the third digit, and the sum of the third and fourth digits is equal to the two more than the fifth digit. How many elements are in the set S? (Note that the leading digit cannot be a zero).

### Five Digit Number

The sum of the digits of a three digit number is eighteen. The first digit is three more than the last digit. There is a repeated digit in the number. What are all possible values of the number?

### Three Digits with Difference

I’m a three digit number, and the sum of my digits is 13. My first two digits differ by 3, and my last two digits differ by 5. What numbers could I be?