One afternoon, you start writing some whole numbers on a (very large) sheet of paper. First you write a one-digit number, then a two-digit, and so on, with each number one digit longer than the last. To make this otherwise dull activity more interesting, you decide never to write a number using all the digits of any previously written number (in any order).

It easy to write numbers at first, but it gets harder and harder as you go.

Will you ever be forced to stop writing, and if so how many numbers might you need to write before this happens?

One afternoon, you start writing some whole numbers on a (very large) sheet of paper. First you write a one-digit number, then a two-digit, and so on, with each number one digit longer than the last. Every number is written in base 10.

To make this otherwise dull activity more interesting, you decide never to write a number using all of the digits in any previously written number (in any order). It easy to write numbers at first, but it gets harder and harder as you go.

Will you ever be forced to stop writing, and if so how many numbers might you need to write before this happens?