12. There are 2021 balls in a crate. The balls are numbered from 1 to 2021 . Erica works out the digit sum for each ball. For example, the digit sum of 2021 is 5, since $2+0+2+1=5$.
Erica notes that balls with equal digit sums have the same colour and balls with different digit sums have different colours.
How many different colours of balls are there in the crate?
SOLUTION
028

The largest possible digit sum is that of 1999 which is 28 . The smallest is that of 1 , which is 1 . Each of the $9 \mathrm{~s}$ in 1999 can be replaced by any of $0,1, \ldots, 8$. So all digit sums between 1 and 28 can be achieved. Therefore there are 28 different digit sums and colours.