A new December and a new bunch of puzzles from mscroggs.co.uk.

I actually did this in a slow way and then realized, there was a quick way.

Slow way: how many ways can you do a 1 digit number (1, the number 9), a 2 digit number (8, the numbers 18, 27, 36 etc.), a 3 digit number etc. Add these together: 1+8+… = 256.

Quick way: This is stars and bars like. Imagine we have 9 stars. *********

If we add no bars, this represents the number 9. All the stars end up in the same bin.

If we add 1 bar, this is a 2 digit number. **|******* represents 27, 2 stars in the first bin, 7 in the second.

If we add 8 bars, it represents the 9 digit number 111111111. Every star is alone in a bin.

With 9 stars, there are 8 positions available between them, 8 ways to add 1 bar between 2 stars. For every one of these 8 positions, we make a decision: add a bar, yes or no? 8 choices, each with 2 options, 2⁸ = 256.