Question

All positive integers which can be written as a sum of one or more distinct powers of 3 are written in increasing order. The first four of them are 1, 3, 4, and 9. What is the 81st?

Answer 1

811

Explanation:

The place values of digits in base-3 are all powers of 3. The numbers 1, 3, 4, 9 written in base-3 are ...

1

10

11

100

You will recognize these as successive numbers in base 2. Then the 81st number of interest is written in base-3 the way 81 is written in base-2:

101 0001

It will be the sum 3^6 +3^4 +3^0 = 729 +81 +1 = 811

The 81st number is 811.