## 1480.11 – Repeating Digits in Addition

The Digit Dog visits you in a dream. He wags his numerical tail and licks your face until it is dripping with digital drool. You are allergic to dogs, but you do not die since you are in a dreamscape.

You wake up with several questions in mind:

a. 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = ?

b. 11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99 = ?

c. 111 + 222 + 333 + 444 + 555 + 666 + 777 + 888 + 999 = ?

d. What is going on here? Does this pattern work for 4-digit addends, 5-digit addends, and so on?

Succumb to your compulsion to answer them.

## Solution

You resolve all of the questions:

a. 45

b. 495

c. 4995

d. The first digit is 4; the last is 5. The rest of the digits are 9's and there is one fewer 9 than the number of digits in the numbers being added. This pattern also works for 4- and 5-digit addends. Why?, you wonder.

You and the Digit Dog will cross paths again someday...