# Problem of the Day: 3/3/13

A certain right triangle has all integer lengths. One of the triangle’s legs is 24 inches. What is the sum of all possible lengths of the other leg?

Solution to yesterday’s problem:

Let’s use our favorite method to solve this problem: the blanks method. There will be 10 blanks for 10 digits:

____ ____ ____ ____ ____ ____ ____ ____ ____ ____

The first digit can be any number. The second digit can be any number except the one used in the first digit, and the same principle goes for the third digit.

__10__ __9__ __8__ ____ ____ ____ ____ ____ ____ ____

The fourth and fifth digits can be any number from 0-9.

__10__ __9__ __8__ __10__ __10__ ____ ____ ____ ____ ____

The last five digits, however, can only be one digit: the digit used in its corresponding spot of the palindrome. For instance, the last digit must be the same as the first digit.

__10__ __9__ __8__ __10__ __10__ __1__ __1__ __1__ __1__ __1__

We don’t need to divide by anything because in phone numbers, order matters! Multiplying the numbers together, we find that there are 72,000 phone numbers that fit the bill.